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Find the number of terms of the A.P $- 12, - 9, - 6, . . . . ., 21$ . If $1$ is added to each term of this A.P., then find the sum of all terms of the A.P thus obtained.

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10, 66

12, 66

14, 66

none of these

answer is B.

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Detailed Solution

Given an A.P

- 12, - 9, - 6, . . . . ., 21

.
We have to find the sum of all terms of the A.P if 1 is added to each term.
From given series, we have,
Last term,

a_{n}

= 21
First term,

a

= -12
Common difference, d= 3

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

\Rightarrow 21 = - 12 + (n - 1) 3

\Rightarrow 21 + 12 = (n - 1) 3

\Rightarrow 33 = (n - 1) 3

\Rightarrow 33 = 3 n - 3

\Rightarrow 33 + 3 = 3 n

\Rightarrow 36 = 3 n

\Rightarrow n = \frac{36}{3}

\Rightarrow n = 12

1 is added to the series to give us,
AP,

- 11, - 8, - 5, . . . . . . . . . . . . ., 22

We know,

S_{n} = \frac{n}{2} (a + l)

S_{n} = \frac{12}{2} (- 11 + 22)

\Rightarrow S_{n} = 6 (11)

{\Rightarrow S}_{n} = 66

Therefore, sum of all terms of AP after adding 1,

is 66

.
Hence, option 2 is correct.

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