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Placement Test Practice — Sequences & Series
Placement Test Practice — Sequences & Series
Practice Test — 20 Questions
1. Find $a_{10}$: arithmetic, $a_1 = 5$, $d = 3$.
2. Find $a_7$: geometric, $a_1 = 2$, $r = 4$.
3. $\sum_{k=1}^{6} k$.
4. Arithmetic series: $a_1 = 1$, $a_{10} = 28$. Find $S_{10}$.
5. Geometric series: $a_1 = 3$, $r = 2$, $n = 5$.
6. $\sum_{k=0}^{\infty} (1/4)^k$.
7. Does $\sum 1/k^2$ converge? (State the type.)
8. Divergence test on $\sum \frac{n}{2n+1}$.
9. Ratio test on $\sum \frac{3^n}{n!}$.
10. Write $0.\overline{9}$ as a fraction.
11. First 3 terms of Maclaurin for $e^x$.
12. Radius of convergence: $\sum \frac{x^n}{n!}$.
13. Find $d$: $a_3 = 11$, $a_8 = 31$.
14. Geometric: $a_1 = 500$, $r = 0.8$. Find $S_\infty$.
15. $\sum_{k=1}^{4} (3k - 2)$.
16. Telescoping: $\sum_{k=1}^{n} \left(\frac{1}{k} - \frac{1}{k+1}\right)$.
17. Is $2, -4, 8, -16, \ldots$ geometric?
18. $\sum_{k=1}^{5} k^2$.
19. Approximate $\cos(0.5)$ using 2 terms of Maclaurin.
20. If $\sum a_n$ converges, what must $\lim a_n$ be?
Show Answer Key
1. $5 + 9(3) = 32$
2. $2 \cdot 4^6 = 8192$
3. $21$
4. $S_{10} = 5(1+28) = 145$
5. $3(2^5-1)/(2-1) = 93$
6. $\frac{1}{1-1/4} = 4/3$
7. Yes — it is a $p$-series with $p = 2 > 1$
8. $\lim \frac{n}{2n+1} = 1/2 \neq 0$; diverges
9. $L = \lim \frac{3}{n+1} = 0$; converges
10. $\frac{0.9}{0.9} = 1$
11. $1 + x + x^2/2$
12. $R = \infty$ (converges for all $x$)
13. $d = (31-11)/5 = 4$
14. $500/0.2 = 2500$
15. $1 + 4 + 7 + 10 = 22$
16. $1 - \frac{1}{n+1}$
17. Yes, $r = -2$
18. $1 + 4 + 9 + 16 + 25 = 55$
19. $1 - (0.5)^2/2 = 1 - 0.125 = 0.875$ (actual ≈ 0.8776)
20. $0$ (necessary but not sufficient)