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rosijanka [135]
1 year ago
8

What is the sum of the arithmetic sequence 3, 9, 15..., if there are 24 terms? (5 points)

Mathematics
1 answer:
Vinil7 [7]1 year ago
3 0

There's a fixed difference of 6 between terms (9 - 3 = 6, 15 - 9 = 6, and so on). The first term of the sequence is 3, so the n-th term is

3 + 6(n-1) = 6n - 3

If there are 24 terms in the sum, then the last term is 6×23 - 3 = 135.

Let S be the sum,

S = 3 + 9 + 15 + \cdots + 123 + 129 + 135

Reverse the order of terms:

S = 135 + 129 + 123 + \cdots + 15 + 9 + 3

If we add up the terms in the same positions, we get twice S on the left side, while on the right side we observe that each pair of terms will sum to 138.

S + S = (3 + 135) + (9 + 129) + (15 + 123) + \cdots + (135 + 3)

2S = 138 + 138 + 138 + \cdots + 138

and since there are 24 terms in the sum, the right side is the sum of 24 copies of 138. In other words,

2S = 24 \times 138

and solving for S gives

S = \dfrac{24\times138}2 = \boxed{1656}

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