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Beyond Advanced · Sequences and Series

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1
Positive numbers x<y<zx<y<z form an AP, x+y+z=18x+y+z=18, and x+1,y+4,z+9x+1,y+4,z+9 form a GP. Find xzxz.
Not attempted
A
821−18\sqrt{21}-1
B
821+18\sqrt{21}+1
C
3636
D
37−82137-8\sqrt{21}
2
A sequence has partial sums Sn=n2−9n+cS_n=n^2-9n+c for every integer n≥1n\ge1. Its first three terms form a nonzero GP. Which discrete graph depicts SnS_n for 1≤n≤81\le n\le8? All candidate graphs use the same scales.
Not attempted
A
Discrete partial sums on the shown scales.
Partial sum versus n14.58-24-104-8-1index
B
Discrete partial sums on the shown scales.
Partial sum versus n14.58-24-104-7-7index
C
Discrete partial sums on the shown scales.
Partial sum versus n14.58-24-104-8-8index
D
Discrete partial sums on the shown scales.
Partial sum versus n14.58-24-104-9-9index
3
Nine increasing real numbers a1,…,a9a_1,\ldots,a_9 form an AP, with ∑ak=45\sum a_k=45 and ∑ak2=285\sum a_k^2=285. Among all permutations π\pi of 1,…,91,\ldots,9, find the least value of ∑k=19akaπ(k)\sum_{k=1}^9 a_k a_{\pi(k)}.
Not attempted
A
285285
B
105105
C
165165
D
225225
4
An increasing AP of seven positive terms has first term aa, difference dd, and last term bb. The sums ∑k=161/(akak+1)\sum_{k=1}^{6}1/(a_ka_{k+1}) and ∑k=16(1/ak−1/ak+1)\sum_{k=1}^{6}(1/a_k-1/a_{k+1}) are respectively 3/143/14 and 3/73/7. If a+b=16a+b=16, find a4a_4.
Not attempted
A
1010
B
88
C
72\frac72
D
66
5
For n≥1n\ge1, let un,vnu_n,v_n be real APs satisfying un+vn=nu_n+v_n=n and unvn=An+Bu_nv_n=An+B, with fixed real A,BA,B. Select every statement that must hold.
Not attempted
A
B=−A2B=-A^2
B
Exactly one of the two sequences is constant.
C
A≥0A\ge0
D
Both sequences can be strictly increasing.
6
Let Sn=αn2+βn+γS_n=\alpha n^2+\beta n+\gamma for n≥1n\ge1, S0=0S_0=0, and an=Sn−Sn−1a_n=S_n-S_{n-1}. Assume α≠0\alpha\ne0. Select every correct equivalence or identity.
Not attempted
A
an+an+2=2an+1a_n+a_{n+2}=2a_{n+1} for all n≥2n\ge2.
B
a1+a3−2a2=γa_1+a_3-2a_2=\gamma.
C
a2−a1=2α+γa_2-a_1=2\alpha+\gamma.
D
(an)(a_n) is an AP if and only if γ=0\gamma=0.
7
An AP an=a+(n−1)da_n=a+(n-1)d has nonzero terms for every positive integer nn. Define bn=1/anb_n=1/a_n and cn=anan+1c_n=a_na_{n+1}. Select every true statement.
Not attempted
A
(cn)(c_n) is an AP if and only if d=0d=0.
B
(bn)(b_n) is a GP if and only if d=0d=0.
C
(bn)(b_n) is an HP.
D
(bn)(b_n) is an AP if and only if d=0d=0.
8
An increasing AP of positive integers has ten terms, sum 200200, and sum of squared distances from its arithmetic mean equal to 330330. How many of its terms are divisible by 33 but not by 44?
Not attempted
Your answer: —
Correct answer: 3
9
The partial sums SnS_n of an AP satisfy S4=20S_4=20 and S10=110S_{10}=110. Let mm satisfy am=30a_m=30. Find ∑k=1m(ak−am+1−k)2\sum_{k=1}^{m}(a_k-a_{m+1-k})^2.
Not attempted
Your answer: —
Correct answer: 4480
10
Two positive numbers have arithmetic mean 55 and geometric mean 33. Insert mm arithmetic means between them. The sum of the squares of these inserted means is 203203. Find the sum of the inserted means.
Not attempted
Your answer: —
Correct answer: 35
11
A real infinite GP with first term a≠0a\ne0 and ratio rr has sum 66, while the sum of squares of its terms is 6060. Find ∑n=1∞narn−1\sum_{n=1}^{\infty}n ar^{n-1}.
Not attempted
A
3030
B
152\frac{15}{2}
C
245\frac{24}{5}
D
365\frac{36}{5}
12
Nine positive increasing GP terms gkg_k satisfy ∏k=19gk=29\prod_{k=1}^9g_k=2^9 and ∑k=19(log⁡2gk)2=69\sum_{k=1}^9(\log_2g_k)^2=69. Which graph depicts log⁡2gk\log_2g_k against kk? All candidates have the same scales.
Not attempted
A
Logarithms at integer indices.
log₂ gₖ versus k159-416-26index
B
Logarithms at integer indices.
log₂ gₖ versus k159-416-35index
C
Logarithms at integer indices.
log₂ gₖ versus k159-416-13index
D
Logarithms at integer indices.
log₂ gₖ versus k159-416-3.805.80index
13
Let g1,…,g12g_1,\ldots,g_{12} be positive GP terms with ratio r>1r>1. How many unordered selections of three distinct terms have product g63g_6^3?
Not attempted
A
1515
B
1717
C
9090
D
1818
14
A positive increasing GP (gn)(g_n) satisfies g1+g2+g3=13g_1+g_2+g_3=13 and g1g2g3=27g_1g_2g_3=27. Evaluate ∑n=1∞gn+1−gn(1+gn)(1+gn+1)\sum_{n=1}^{\infty}\frac{g_{n+1}-g_n}{(1+g_n)(1+g_{n+1})}.
Not attempted
A
34\frac34
B
13\frac13
C
11
D
12\frac12
15
A nonzero real GP has S2=3S_2=3 and S4=15S_4=15, where SmS_m is its first-mm-term sum. Select every statement forced by these data.
Not attempted
Allowed first two indices: i < j; k > jij112233445566778899101011111212
A
a>0a>0
B
g1g2g3=8g_1g_2g_3=8
C
S6=63S_6=63
D
The GP has no finite infinite sum.
16
For a positive nonconstant GP g1,…,g2m+1g_1,\ldots,g_{2m+1} with m≥1m\ge1, define Lk=ln⁡gkL_k=\ln g_k and M=gm+1M=g_{m+1}. Select every assertion valid for every positive ratio other than 11.
Not attempted
A
∑k=12m+1ln⁡(1/gk)>0\sum_{k=1}^{2m+1}\ln(1/g_k)>0.
B
(Lk)(L_k) is an AP.
C
∏k=12m+1gk=M2m+1\prod_{k=1}^{2m+1}g_k=M^{2m+1}.
D
∑k=12m+1gk>(2m+1)M\sum_{k=1}^{2m+1}g_k>(2m+1)M.
17
For real x≠−1x\ne-1, consider ∑n=1∞(x−2)((x−1)/(x+1))n−1\sum_{n=1}^{\infty}(x-2)((x-1)/(x+1))^{n-1}. A zero first term makes this the zero series. Select every correct statement.
Not attempted
A
The series converges exactly for x>0x>0.
B
For x>0x>0, its sum is (x−2)(x+1)/2(x-2)(x+1)/2.
C
The series converges at x=0x=0.
D
The sum is positive for every x>0x>0.
18
A real GP gn=arn−1g_n=ar^{n-1} has ∑gn=3\sum g_n=3 and ∑∣gn∣=9\sum |g_n|=9. Find 3∑n=1∞n2gn3\sum_{n=1}^{\infty}n^2g_n.
Not attempted
Endpoint intervals and seven inserted means2b₁…b₇512
Your answer: —
Correct answer: 2
19
Insert mm geometric means between 22 and 512512 to create an increasing positive GP. The product of the inserted means is 2352^{35}. If the means are b1,…,bmb_1,\ldots,b_m, find 256∑j=1m1/bj256\sum_{j=1}^{m}1/b_j.
Not attempted
Your answer: —
Correct answer: 127
20
Let dnd_n be the decimal consisting of nn copies of the block 2727 after the decimal point. Put en=3/11−dne_n=3/11-d_n. Find 1067328960000∑n=1∞n(n+1)en\frac{10673289}{60000}\sum_{n=1}^{\infty}n(n+1)e_n.
Not attempted
Your answer: —
Correct answer: 1
21
Positive numbers x<y<zx<y<z form an HP, with x+y+z=13x+y+z=13 and xyz=72xyz=72. Find x2+z2x^2+z^2.
Not attempted
A
4545
B
8181
C
3636
D
7272
22
Two distinct positive numbers have arithmetic, geometric and harmonic means A,G,HA,G,H. If A−G=2(G−H)A-G=2(G-H) and their sum is 1616, find the square of their difference.
Not attempted
A
00
B
128128
C
240240
D
192192
23
Five positive increasing HP terms are h1,…,h5h_1,\ldots,h_5, with h1=2h_1=2 and h5=10h_5=10. Find ∑k=14khkhk+1\sum_{k=1}^{4}k h_kh_{k+1}.
Not attempted
HP terms; weights belong to adjacent linksh₁h₂h₃h₄h₅1234
A
7703\frac{770}{3}
B
400400
C
8153\frac{815}{3}
D
8080
24
Positive numbers a<ca<c have arithmetic mean 55 and harmonic mean 9/59/5. Let a,b,ca,b,c be an HP, and set u=a/(b+c),v=b/(a+c),w=c/(a+b)u=a/(b+c),v=b/(a+c),w=c/(a+b). Find 1/u+1/v+1/w1/u+1/v+1/w.
Not attempted
A
103\frac{10}{3}
B
503\frac{50}{3}
C
593\frac{59}{3}
D
1515
25
For two distinct positive numbers, their means satisfy A−G=λ(G−H)A-G=\lambda(G-H). Select every correct assertion.
Not attempted
A
λ>1\lambda>1
B
A/G=G/H=λA/G=G/H=\lambda
C
If λ=3\lambda=3, the larger-to-smaller endpoint ratio is 17+12217+12\sqrt2.
D
λ\lambda uniquely determines the difference between the two numbers.
26
Distinct positive numbers a,b,ca,b,c are in HP. Put T=a+b+cT=a+b+c. Select every true statement.
Not attempted
A
a2,b2,c2a^2,b^2,c^2 form an HP.
B
1/(b−a)+1/(b−c)=2/b1/(b-a)+1/(b-c)=2/b.
C
a+c=2ba+c=2b
D
a/(b+c),b/(a+c),c/(a+b)a/(b+c),b/(a+c),c/(a+b) form an HP.
27
Let hn=1/(αn+β)h_n=1/(\alpha n+\beta), with α>0\alpha>0 and α+β>0\alpha+\beta>0. Select every true statement about the indicated ordinary series.
Not attempted
A
∑j=1∞h2j\sum_{j=1}^{\infty}h_{2^j} converges.
B
lim⁡n→∞hn=0\lim_{n\to\infty}h_n=0.
C
∑n=1∞hnhn+1=1/[α(α+β)]\sum_{n=1}^{\infty}h_nh_{n+1}=1/[\alpha(\alpha+\beta)].
D
∑n=1∞hn\sum_{n=1}^{\infty}h_n converges.
28
Insert mm harmonic means h1,…,hmh_1,\ldots,h_m between 11 and 1/51/5. Their reciprocal square sum is ∑j=1m1/hj2=70\sum_{j=1}^m1/h_j^2=70. Find mm.
Not attempted
Harmonic means: reciprocals divide the interval1h₁…hₘ1/5
Your answer: —
Correct answer: 7
29
Positive real numbers x,yx,y satisfy x+y=18x+y=18 and 1/x+1/y=1/21/x+1/y=1/2. For the three-term HP x,h,yx,h,y, find 2(x−h)2+2(y−h)22(x-h)^2+2(y-h)^2.
Not attempted
Your answer: —
Correct answer: 280
30
A decreasing positive HP satisfies h1=3h_1=3 and ∑n=1∞hnhn+1=18\sum_{n=1}^{\infty}h_nh_{n+1}=18. Find the least positive integer NN such that ∑n=1Nhnhn+1≥17\sum_{n=1}^{N}h_nh_{n+1}\ge17.
Not attempted
Your answer: —
Correct answer: 34
31
For x≥2x\ge2, put y=1/xy=1/x and f(x)=x2+y2+4/(x+y)2f(x)=x^2+y^2+4/(x+y)^2. Which graph correctly depicts f(x)f(x) on 2≤x≤42\le x\le4? All candidates use the same scales.
Not attempted
A
Objective value on the feasible interval.
Objective value versus x234310.5184.2516.06x
B
Objective value on the feasible interval.
Objective value versus x234310.518314.25x
C
Objective value on the feasible interval.
Objective value versus x234310.5184.8916.28x
D
Objective value on the feasible interval.
Objective value versus x234310.5184.8912.89x
32
Positive numbers a<b<ca<b<c form an AP, have sum 66, and satisfy c≥2ac\ge2a. Find the greatest value of abc/(a2+c2)abc/(a^2+c^2).
Not attempted
A
23\frac23
B
45\frac45
C
11
D
89\frac89
33
Positive a,b,ca,b,c satisfy abc=1abc=1 and a=2ba=2b. Find the least value of (a+b+c)(1/a+1/b+1/c)(a+b+c)(1/a+1/b+1/c).
Not attempted
A
112+32\frac{11}{2}+3\sqrt2
B
99
C
252\frac{25}{2}
D
112+23\frac{11}{2}+2\sqrt3
34
Five positive GP terms g1,…,g5g_1,\ldots,g_5 have product 11 and g2+g4≥3g_2+g_4\ge3. Find the minimum of ∑k=15gk2\sum_{k=1}^{5}g_k^2.
Not attempted
A
55
B
4747
C
5757
D
5555
35
Positive a,b,ca,b,c satisfy a+b+c=1a+b+c=1 and a≥1/2a\ge1/2. Select every sharp bound below.
Not attempted
A
max⁡(ab+ac+bc)=5/16\max(ab+ac+bc)=5/16.
B
max⁡abc=1/27\max abc=1/27.
C
max⁡abc=1/32\max abc=1/32.
D
min⁡(1/a+1/b+1/c)=10\min(1/a+1/b+1/c)=10.
36
Positive x,yx,y satisfy x+y=6x+y=6 and x≥5x\ge5. Select every correct extremal assertion.
Not attempted
A
x=yx=y is feasible.
B
max⁡x2y=25\max x^2y=25.
C
max⁡xy=9\max xy=9.
D
min⁡(1/x+1/y)=2/3\min(1/x+1/y)=2/3.
37
Positive numbers u,vu,v have uv=4uv=4 and u+v≤5u+v\le5. Let A,G,HA,G,H be their means. Select every sharp bound or identity.
Not attempted
A
max⁡(A−H)=9/10\max(A-H)=9/10.
B
max⁡(u−v)2=9\max(u-v)^2=9.
C
AH=4AH=4
D
min⁡H=2\min H=2.
38
Positive x,y,zx,y,z satisfy xyz=64xyz=64 and x=2yx=2y. Let MM be the minimum possible value of x+y+zx+y+z. Find M3M^3.
Not attempted
Your answer: —
Correct answer: 1944
39
Seven positive terms form a GP with product 11. If the sum of its third and fifth terms is at least 44, find the least possible sum of all seven terms.
Not attempted
Your answer: —
Correct answer: 71
40
Four positive numbers a1,a2,a3,a4a_1,a_2,a_3,a_4 have product 11 and satisfy a1=4a2a_1=4a_2. If MM is the least possible value of a1+a2+a3+a4a_1+a_2+a_3+a_4, find M2M^2.
Not attempted
Your answer: —
Correct answer: 20
41
For un=(a+nd)rnu_n=(a+nd)r^n, n≥0n\ge0, assume d≠0d\ne0 and 0<r<10<r<1. If ∑un=0\sum u_n=0, ∑nun=6\sum n u_n=6 and ∑n2un=42\sum n^2u_n=42, find rr.
Not attempted
A
35\frac35
B
23\frac23
C
12\frac12
D
34\frac34
42
Find ∑1≤i<j<∞(ij+i+j)/2i+j\sum_{1\le i<j<\infty}(ij+i+j)/2^{i+j}.
Not attempted
Strict upper triangle; the infinite grid continuesij1122334455667788
A
44
B
17227\frac{172}{27}
C
4427\frac{44}{27}
D
8627\frac{86}{27}
43
An AP (an)(a_n) satisfies ∑n=1∞an/2n=0\sum_{n=1}^{\infty}a_n/2^n=0 and ∑n=1∞an/3n=1\sum_{n=1}^{\infty}a_n/3^n=1. Find ∑n=1∞nan/2n\sum_{n=1}^{\infty}n a_n/2^n.
Not attempted
A
00
B
−16-16
C
−8-8
D
88
44
For every N≥1N\ge1, define RN=∑n=N+1∞(2n−1)/2nR_N=\sum_{n=N+1}^{\infty}(2n-1)/2^n. Find ∑N=1∞RN/2N\sum_{N=1}^{\infty}R_N/2^N.
Not attempted
A
119\frac{11}{9}
B
179\frac{17}{9}
C
239\frac{23}{9}
D
173\frac{17}{3}
45
Let FN(r)=∑n=1Nnrn−1F_N(r)=\sum_{n=1}^{N}nr^{n-1} for real rr. Select every correct statement.
Not attempted
Shifted finite sum includes the final power1r…rᴺ⁻¹rᴺ
A
For r≠1r\ne1, FN(r)=[1−(N+1)rN+NrN+1]/(1−r)2F_N(r)=[1-(N+1)r^N+Nr^{N+1}]/(1-r)^2.
B
FN(1)=N(N+1)/2F_N(1)=N(N+1)/2.
C
F2m(−1)=−mF_{2m}(-1)=-m.
D
For ∣r∣<1|r|<1, lim⁡FN(r)=1/(1−r)2\lim F_N(r)=1/(1-r)^2.
46
Let T=∑n=1∞n2/2nT=\sum_{n=1}^{\infty}n^2/2^n and O=∑n≥1, n oddn2/2nO=\sum_{n\ge1,\ n\text{ odd}}n^2/2^n. Select every correct assertion.
Not attempted
A
∑n=1∞(−1)n−1n2/2n=2/27\sum_{n=1}^{\infty}(-1)^{n-1}n^2/2^n=2/27.
B
O=T/2O=T/2
C
The even-index part is T/4T/4.
D
O=82/27O=82/27
47
For real a,da,d, consider ∑n=0∞(a+nd)rn\sum_{n=0}^{\infty}(a+nd)r^n, where rr is real. Select every correct assertion.
Not attempted
A
The series always diverges when ∣r∣≥1|r|\ge1.
B
If a=0a=0, the series converges for every rr.
C
For ∣r∣<1|r|<1, the sum is a/(1−r)+dr/(1−r)2a/(1-r)+dr/(1-r)^2.
D
At r=−1r=-1, convergence holds if and only if a=d=0a=d=0.
48
For positive integers NN, let SN=∑n=1N(n−4)/2nS_N=\sum_{n=1}^{N}(n-4)/2^n. Find ∑N=1∞(SN+2)\sum_{N=1}^{\infty}(S_N+2).
Not attempted
Your answer: —
Correct answer: 0
49
Evaluate ∑n=1∞2−n∑k=1nk(n+1−k)\sum_{n=1}^{\infty}2^{-n}\sum_{k=1}^{n}k(n+1-k).
Not attempted
Inner convolution indices: 1 ≤ k ≤ nij112233445566
Your answer: —
Correct answer: 8
50
A sequence is defined by bn=2n∑k=n+1∞k2/2kb_n=2^n\sum_{k=n+1}^{\infty}k^2/2^k. Find b12−2b11+b10b_{12}-2b_{11}+b_{10}.
Not attempted
Your answer: —
Correct answer: 2
51
A sequence satisfies T1=2T_1=2 and Tn+1−Tn=n(n+1)T_{n+1}-T_n=n(n+1) for n≥1n\ge1. Find ∑k=110[(T2k−T2k−1)−4k(k+1)]+∑k=110(Tk+1−Tk)\sum_{k=1}^{10}[(T_{2k}-T_{2k-1})-4k(k+1)]+\sum_{k=1}^{10}(T_{k+1}-T_k).
Not attempted
A
330330
B
220220
C
110110
D
14301430
52
Eleven increasing AP terms have sum 3333 and square sum 209209. Find ∑1≤i<j≤11(ai−aj)2\sum_{1\le i<j\le11}(a_i-a_j)^2.
Not attempted
Each unordered index pair occurs onceij11223344556677889910101111
A
20902090
B
12101210
C
24202420
D
110110
53
Find ∑1≤i<j<k≤8ijk\sum_{1\le i<j<k\le8}ijk.
Not attempted
First two indices i < j; third index k > jij1122334455667788
A
45364536
B
77767776
C
41044104
D
60486048
54
Evaluate ∑1≤i<j≤8i(9−j)\sum_{1\le i<j\le8}i(9-j).
Not attempted
The summation region: i < jij1122334455667788
A
420420
B
12961296
C
252252
D
210210
55
For integer n≥2n\ge2, define Pn=∑1≤i<j≤nijP_n=\sum_{1\le i<j\le n}ij and Dn=∑1≤i<j≤n(j−i)D_n=\sum_{1\le i<j\le n}(j-i). Select every correct identity.
Not attempted
Gap d = j − i; strict upper triangleij1122334455667788
A
∑i<j(j−i)2=n2(n2−1)/12\sum_{i<j}(j-i)^2=n^2(n^2-1)/12.
B
Dn=n(n−1)/2D_n=n(n-1)/2 for every nn.
C
Pn=n(n+1)(n−1)(3n+2)/24P_n=n(n+1)(n-1)(3n+2)/24.
D
Dn=n(n2−1)/6D_n=n(n^2-1)/6.
56
A sequence obeys T1=1T_1=1, T2=4T_2=4 and Tn+2−2Tn+1+Tn=2T_{n+2}-2T_{n+1}+T_n=2 for all n≥1n\ge1. Select every true statement.
Not attempted
A
The partial-sum polynomial has degree two.
B
∑k=1nTk=n(n+1)(2n+1)/6\sum_{k=1}^{n}T_k=n(n+1)(2n+1)/6.
C
Tn+1−Tn=2n+1T_{n+1}-T_n=2n+1.
D
(Tn)(T_n) is an AP.
57
For n≥2n\ge2, let A=∑k=1nkA=\sum_{k=1}^{n}k and B=∑k=1nk2B=\sum_{k=1}^{n}k^2. Select every correct statement.
Not attempted
Full square: ordered pairs, including diagonalij11223344556677
A
∑i=1n∑j=1nij=A2\sum_{i=1}^n\sum_{j=1}^nij=A^2.
B
∑i=1n∑j=1n(i+j)2=2nB+2A2\sum_{i=1}^n\sum_{j=1}^n(i+j)^2=2nB+2A^2.
C
∑i<j(i2+j2)=(n−1)B\sum_{i<j}(i^2+j^2)=(n-1)B.
D
B=A2B=A^2
58
Let ak=∑j=1kja_k=\sum_{j=1}^{k}j and bk=∑j=k10jb_k=\sum_{j=k}^{10}j, with b11=0b_{11}=0. Find ∑k=110(ak+bk+1)(bk+1−ak)\sum_{k=1}^{10}(a_k+b_{k+1})(b_{k+1}-a_k).
Not attempted
Two disjoint ranges at the cut1…kk+110aₖbₖ₊₁
Your answer: —
Correct answer: 6050
59
For n≥1n\ge1, let Tn=∑k=1nk(k+1)T_n=\sum_{k=1}^{n}k(k+1). Find ∑n=16(Tn−Tn−1)(7−n)\sum_{n=1}^{6}(T_n-T_{n-1})(7-n), where T0=0T_0=0.
Not attempted
Nested sum indices before differencingij112233445566
Your answer: —
Correct answer: 252
60
A quadratic polynomial pp satisfies ∑k=1np(k)=n(n+1)(2n+7)\sum_{k=1}^{n}p(k)=n(n+1)(2n+7) for every positive integer nn. Find ∑k=18p(2k−1)\sum_{k=1}^{8}p(2k-1).
Not attempted
Your answer: —
Correct answer: 4848
61
A positive increasing AP (an)(a_n) satisfies ∑n≥11/(anan+1)=1/12\sum_{n\ge1}1/(a_na_{n+1})=1/12 and ∑n≥1(1/an−1/an+1)=1/4\sum_{n\ge1}(1/a_n-1/a_{n+1})=1/4. Find ∑n≥11/(anan+1an+2)\sum_{n\ge1}1/(a_na_{n+1}a_{n+2}).
Not attempted
A
1168\frac1{168}
B
184\frac1{84}
C
148\frac1{48}
D
128\frac1{28}
62
Using principal inverse tangents, evaluate ∑k=−33tan⁡−1(1/(k2+k+1))\sum_{k=-3}^{3}\tan^{-1}(1/(k^2+k+1)).
Not attempted
A
−tan⁡−1(7/11)-\tan^{-1}(7/11)
B
tan⁡−1(7/11)\tan^{-1}(7/11)
C
π/2+tan⁡−1(7/11)\pi/2+\tan^{-1}(7/11)
D
π−tan⁡−1(7/11)\pi-\tan^{-1}(7/11)
63
Let CN=∑k=1N1/(k+2+k)C_N=\sum_{k=1}^{N}1/(\sqrt{k+2}+\sqrt{k}). Find lim⁡N→∞N[CN−N+(1+2)/2]\lim_{N\to\infty}\sqrt N[C_N-\sqrt N+(1+\sqrt2)/2].
Not attempted
A
32\frac32
B
12\frac12
C
34\frac34
D
11
64
Let x1=3x_1=3 and xn+1=xn/(1+2xn)x_{n+1}=x_n/(1+2x_n) for n≥1n\ge1. Evaluate ∑n=1∞xnxn+1xn+2\sum_{n=1}^{\infty}x_nx_{n+1}x_{n+2}.
Not attempted
A
67\frac67
B
928\frac9{28}
C
914\frac9{14}
D
32\frac32
65
Let AN=∑k=1N1/[k(k+2)]A_N=\sum_{k=1}^{N}1/[k(k+2)] and BN=∑k=1Nk/[k(k+2)]B_N=\sum_{k=1}^{N}k/[k(k+2)]. Select every correct statement.
Not attempted
Gap-two cancellation: two ends survive12…N+1N+2
A
AN=3/4−1/[2(N+1)]−1/[2(N+2)]A_N=3/4-1/[2(N+1)]-1/[2(N+2)].
B
lim⁡N(3/4−AN)=1\lim N(3/4-A_N)=1.
C
BN→3/4B_N\to3/4
D
AN=1/2−1/[2(N+2)]A_N=1/2-1/[2(N+2)].
66
Let LN=∑k=1Nln⁡{k(k+3)/[(k+1)(k+2)]}L_N=\sum_{k=1}^{N}\ln\{k(k+3)/[(k+1)(k+2)]\}. Select every true assertion.
Not attempted
A
(LN)(L_N) is strictly decreasing.
B
lim⁡N(LN+ln⁡3)=2\lim N(L_N+\ln3)=2.
C
lim⁡LN=−ln⁡2\lim L_N=-\ln2.
D
lim⁡LN=−ln⁡3\lim L_N=-\ln3.
67
Let y1=t>0y_1=t>0 and yn+1=yn/(1+yn)y_{n+1}=y_n/(1+y_n). Select every true assertion.
Not attempted
A
∑n≥1ynyn+2=t(t+2)/[2(t+1)]\sum_{n\ge1}y_ny_{n+2}=t(t+2)/[2(t+1)].
B
∑n≥1yn\sum_{n\ge1}y_n converges.
C
yn=t/[1+(n−1)t]y_n=t/[1+(n-1)t].
D
∑n≥1ynyn+1=t\sum_{n\ge1}y_ny_{n+1}=t.
68
For PN=∏k=3N(1−4/k2)P_N=\prod_{k=3}^{N}(1-4/k^2), find the least integer N≥3N\ge3 for which PN<1/5P_N<1/5.
Not attempted
Your answer: —
Correct answer: 22
69
Evaluate 120∑n=1∞n+2n(n+1)(n+3)(n+4)120\sum_{n=1}^{\infty}\frac{n+2}{n(n+1)(n+3)(n+4)}.
Not attempted
Your answer: —
Correct answer: 15
70
Let x=tan⁡−1(1/3)x=\tan^{-1}(1/3) in its principal range. Evaluate 7∑k=03sin⁡xcos⁡(kx)cos⁡((k+1)x)7\sum_{k=0}^{3}\frac{\sin x}{\cos(kx)\cos((k+1)x)}.
Not attempted
Your answer: —
Correct answer: 24
Convert each term into a tangent difference, then apply the double-angle formula twice.
The identity tan⁡((k+1)x)−tan⁡(kx)=sin⁡x/[cos⁡(kx)cos⁡((k+1)x)]\tan((k+1)x)-\tan(kx)=\sin x/[\cos(kx)\cos((k+1)x)] turns the sum into tan⁡4x\tan4x. From tan⁡x=1/3\tan x=1/3, tan⁡2x=(2/3)/(1−1/9)=3/4\tan2x=(2/3)/(1-1/9)=3/4. Thus tan⁡4x=(3/2)/(1−9/16)=24/7\tan4x=(3/2)/(1-9/16)=24/7. The denominators are defined: 0<x<π/80<x<\pi/8, so 0≤kx≤4x<π/20\le kx\le4x<\pi/2. This also validates the principal-angle choice. Multiplying by 77 gives 2424. Replacing the difference by a sum, or applying the double-angle rule without its denominator, gives an incorrect value.