Arithmetic series

Front

What is an arithmetic series?

Back

An arithmetic series is the sum of the terms of an arithmetic sequence, where each term after the first is obtained by adding a constant difference to the previous term.

Front

What is the formula for the sum of the first n terms of an arithmetic series?

Back

The formula is: S_n = \frac{n}{2} \times (2a_1 + (n-1)d) or S_n = \frac{n}{2} \times (a_1 + a_n), where a_1 is the first term, a_n is the nth term, d is the common difference, and n is the number of terms.

Front

If a_1 = 5 and d = 3, what is the 10th term of the arithmetic sequence?

Back

The 10th term a_{10} = a_1 + (10-1)d = 5 + 9 \times 3 = 32.

Front

Calculate the sum of the first 10 terms of the series: 2 + 5 + 8 + ...

Back

S_{10} = \frac{10}{2} \times (2 + 29) = 5 \times 31 = 155.

Front

What is the common difference in the arithmetic series: 4, 7, 10, 13, ...?

Back

The common difference d = 7 - 4 = 3.

Front

Given a_1 = -3, d = 2, find S_n if n = 7.

Back

S_7 = \frac{7}{2} \times (2 \times -3 + (7-1) \times 2) = \frac{7}{2} \times (-6 + 12) = \frac{7}{2} \times 6 = 21.

Front

What is the 5th term of the arithmetic sequence where a_1 = 10 and d = -2?

Back

a_5 = a_1 + (5-1)d = 10 + 4 \times (-2) = 10 - 8 = 2.

Front

If S_n = 100, a_1 = 10, and d = 5, how many terms are in the series?

Back

Using S_n = \frac{n}{2} \times (2a_1 + (n-1)d), we solve for n: 100 = \frac{n}{2} \times (20 + (n-1) \times 5).

Front

Evaluate the series: 1 + 4 + 7 + ... + 28.

Back

This is an arithmetic series with a_1 = 1, d = 3, and n = 10. S_{10} = \frac{10}{2} \times (1 + 28) = 5 \times 29 = 145.

Front

What is the formula to find the nth term of an arithmetic sequence?

Back

The nth term a_n = a_1 + (n-1)d.