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Arithmetic Mean

Let a1,a2,a3,...

Question

Let $a_{1}, a_{2}, a_{3}, \dots$ be an A.P. such that $a_{6} + a_{9} + a_{12} + a_{15} = 20$ . The value of sum of the first $20$ terms is equal to

100

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150

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140

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200

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Solution

The correct option is A $100$
Let the first term and common different of the A.P. is $a_{1}$ and $d$
so $a_{6} + a_{9} + a_{12} + a_{15} = 20$
$\Rightarrow a_{1} + 5 d + a_{1} + 8 d + a_{1} + 11 d + a_{1} + 14 d = 20 \Rightarrow 2 a_{1} + 2 (a_{1} + 19 d) = 20 \Rightarrow a_{1} + a_{20} = 10$
Now, $S_{20} = \frac{20}{2} [a_{1} + a_{20}] = 100$

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