Sequences Module - Practice Flashcards (Question and Answer)

40 flashcards covering definitions, formulas, and classic examples from the sequences module (arithmetic, geometric, harmonic, Fibonacci).

What is an arithmetic sequence?

A sequence with a constant difference between consecutive terms, known as the common difference.

What is a geometric sequence?

A sequence where each term is obtained by multiplying the previous term by a constant ratio.

What is a harmonic sequence?

A sequence whose reciprocals form an arithmetic sequence.

What is a Fibonacci sequence?

A sequence where each term is the sum of the two preceding terms.

What is the formula for the nth term of an arithmetic sequence?

an = a1 + (n − 1)d.

What is the formula for the nth term of a geometric sequence?

an = a1 r^(n − 1).

What is the sum of the first n terms of an arithmetic sequence?

Sn = n/2 [2a1 + (n − 1)d] (or Sn = n/2 (a1 + a_n)).

What is the sum of the first n terms of a geometric sequence (r ≠ 1)?

Sn = a1 (1 − r^n) / (1 − r).

What is the sum to infinity of a geometric sequence?

Sinfty = a1 / (1 − r) provided |r| < 1.

When does a geometric series have a finite sum to infinity?

When the common ratio satisfies |r| < 1.

What is the next term in the arithmetic sequence 4, 7, 10, 13, …?

16.

What is the next term in the geometric sequence 3, 6, 12, 24, …?

48.

What is the next term in the Fibonacci sequence 1, 1, 2, 3, 5, 8, …?

13.

What is the sum of the first 10 terms of the arithmetic sequence 5, 9, 13, 17, …?

230.

What is the sum of the integers from 1 to 100?

5050.

What is the 10th term of the arithmetic sequence 5, 12, 19, 26, …?

68.

If an arithmetic sequence starts at a1 = −3 with d = 2 and ends at 23, how many terms are there?

14 terms.

Is the sequence 3, 7, 11, 15, 19 an arithmetic sequence? If yes, what is its common difference?

Yes; common difference is 4.

Is the sequence 4, 16, 64, 256 a geometric sequence? If yes, what is its common ratio?

Yes; common ratio is 4.

What is the nth term of the sequence 7, 9, 11, 13, …?

a_n = 7 + 2(n − 1) = 2n + 5.

What is the general form for the nth term of a harmonic sequence?

If the reciprocals form an AP with first term p and difference q, then a_n = 1/(p + (n−1)q).

What is the sum of the first 7 terms of a geometric sequence with a_1 = 3 and r = 5?

58,593.

Find the sum to infinity of the geometric sequence 64, 16, 4, 1, …

256/3.

In a geometric sequence with r = −1, what is the sum of the first n terms?

If n is even, the sum is 0; if n is odd, the sum equals a_1.

What method did Gauss reportedly use to sum the integers from 1 to 100?

Pair numbers (1+100), (2+99), …, giving 50 pairs each summing to 101; total 5050.

What is the nth term of the arithmetic sequence 3, 4, 5, 6, 7, …?

a_n = n + 2.

What is the sum of the first 4 terms of the arithmetic sequence 1, 4, 7, 10?

22.

What is the sum of the first 25 multiples of 8?

2600.

What is the sum of the first 5 terms of the geometric sequence 3, −3, 3, −3, 3?

3.

What is the sum of the first 6 terms of the geometric sequence 2, 4, 8, 16, 32, 64?

126.

What is the nth term of the sequence 6, 5, 4, 3, 2, 1?

a_n = 7 − n.

What is the sum of the first 10 terms of the arithmetic sequence 1, 3, 5, 7, 9, 11, 13, 15, 17, 19?

100.

What is the common ratio of the geometric sequence −5, 10, −20, 40?

−2.

What is the sum to infinity of the sequence 1, 1/2, 1/4, 1/8, …?

2.

Is the sequence 1, 1/2, 1/3, 1/4 a harmonic sequence? Why?

Yes, because its reciprocals form an arithmetic sequence.

What is the next term in the Fibonacci sequence after 8?

13.

If a1 = 8 and d = 3, what is a5?

20.

If a5 = 20 and d = 3, what is a1?

8.

If a1 = 8 and a3 = 32 for a geometric sequence, what are the possible common ratios?

r^2 = 32/8 = 4, so r = 2 or r = −2.

What is the standard formula for the sum of the first n terms of a geometric sequence?

Sn = a1 (1 − r^n) / (1 − r) for r ≠ 1; if r = 1, Sn = n a1.