Sequences & Series Flashcards

Which formula gives the sum of the first n natural numbers?

Answer: n(n+1)/2

The sum 1 + 2 + 3 + ... + n equals n(n+1)/2, a standard result.

The sum to infinity of a geometric series is 12 and the first term is 4. What is the common ratio?

Answer: 2/3

Using S∞ = a/(1−r): 12 = 4/(1−r) gives 1−r = 1/3, so r = 2/3.

Find the 15th term of the arithmetic sequence: 7, 10, 13, 16, ...

Answer: 49

T_15 = 7 + (14)(3) = 7 + 42 = 49.

Which of the following infinite geometric series converges?

Answer: 5 + 5/2 + 5/4 + ...

The series 5 + 5/2 + 5/4 + ... has ratio r = 1/2, and |r| < 1 guarantees convergence.

If the sum ∑_{k=1}^{n} k = 210, what is the value of n?

Answer: 20

Using n(n+1)/2 = 210 gives n(n+1) = 420; since 20 × 21 = 420, n = 20.

A sequence is defined by a₁ = 2 and a_{n+1} = 3a_n − 1. What is the value of a₄?

Answer: 41

a₂ = 5, a₃ = 14, a₄ = 3(14) − 1 = 41.

The sum of the first 5 terms of a geometric sequence with common ratio 2 is 31. What is the first term?

Answer: 1

S₅ = a(2⁵ − 1)/(2 − 1) = 31a = 31, so a = 1.