Expert Answer to "What is the sum of the arithmetic sequence..."

Wendy Larsen

Answered question

2023-01-05

What is the sum of the arithmetic sequence 152, 138, 124 if there are 24 terms?

Myncboyncgm6

Beginner2023-01-06Added 8 answers

An arithmetic sequence's general term can be expressed in the following way:

a_{n} = a + d (n - 1)

where d is the common difference and a is the initial term.
Therefore:

2 \sum_{n = 1}^{N} a_{n} = \sum_{n = 1}^{N} (a_{n} + a_{N + 1 - n})

2 \sum_{n = 1}^{N} a_{n} = \sum_{n = 1}^{N} (a + d (n - 1) + a + d (N - n))

2 \sum_{n = 1}^{N} a_{n} = \sum_{n = 1}^{N} (2 a + d N - d)

2 \sum_{n = 1}^{N} a_{n} = N (2 a + d N - d)

So:

s_{N} = \sum_{n = 1}^{N} a_{n} = \frac{1}{2} N (2 a + d N - d)

Equivalently, the sum of the first #N# terms of an arithmetic sequence is the number of terms multiplied by the average term.
Then:

s_{N} = N \cdot \frac{a_{1} + a_{N}}{2}

s_{N} = \frac{1}{2} N (a + d (1 - 1) + a + d (N - 1))

s_{N} = \frac{1}{2} N (2 a + d N - d)

For our example:

a = 152

d = 138 - 152 = - 14

N = 24

So:

s_{24} = \frac{1}{2} (24) (2 (152) + (- 14) (24) - - 14)

s_{24} = 12 (304 - 336 + 14)

s_{24} = 12 (- 18)

s_{24} = - 216

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