Evaluate
$82 + 72 + 62 + . . . .$ to $12$ terms.

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Solution

Step 1. Note the given data:
The given series $82 + 72 + 62 + . . . .$ to $12$ terms is an A.P. with
First term $a = 82$
Number of terms $n = 12$
Common term $72 - 82 = - 10$
Step 2. Find the sum of the first $12$ terms of the given A.P.
The formula of the sum of the first $n$ terms of an A.P. is $S_{n} = \frac{n}{2} (2 a + (n - 1) d)$
Where $a$ is the first term and $d$ is common difference.
Here $n = 12, a = 82, d = - 10$
Substituting the value
$S_{12} = \frac{12}{2} (2 (82) + (12 - 1) (- 10)) S_{12} = 6 (164 + (11) (- 10)) S_{12} = 6 (164 - 110) S_{12} = 6 (54) S_{12} = 324$
Hence, the sum of first $12$ terms of the given A.P. is $324$ .

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