Sequences & Series Flashcards

The nth term of a sequence is T_n = 3n − 2. For what value of n does T_n = 46?

Answer: 16

Setting 3n − 2 = 46 gives 3n = 48, so n = 16.

The sum of the first n terms of a series is S_n = 2n² + 3n. What is the 5th term?

Answer: 21

T₅ = S₅ − S₄ = (50+15) − (32+12) = 65 − 44 = 21.

Three consecutive terms of an arithmetic sequence sum to 30 and their product is 750. What is the middle term?

Answer: 10

Let the terms be a−d, a, a+d; their sum 3a = 30 gives a = 10, and 10(100−d²) = 750 gives d = ±5, confirming the middle term is 10.

The first three terms of a geometric sequence are x+2, 3x, and 5x+4. What is the common ratio?

Answer: 2

Setting (3x)² = (x+2)(5x+4) gives x = 4, yielding the terms 6, 12, 24 with ratio 2.

What is the sum of the first 8 terms of the geometric series 3 + 6 + 12 + ...?

Answer: 765

S₈ = 3(2⁸ − 1)/(2 − 1) = 3 × 255 = 765.

The nth term of a sequence is a_n = (−1)^n × n²/2. What is the sum of the first 4 terms?

Answer: 5

a₁ = −1/2, a₂ = 2, a₃ = −9/2, a₄ = 8; sum = −1/2 + 2 − 9/2 + 8 = 5.

In a geometric sequence, the sum of the 3rd and 4th terms equals 12 times the first term. What is the common ratio?

Answer: 2

T₃ + T₄ = ar² + ar³ = ar²(1+r) = 12a, so r²(1+r) = 12; r = 2 satisfies this since 4×3 = 12.