Question 15

The common difference of the $$A.P.: a_{1},a_{2},.....,a_{m}$$ is 13 more than the common difference of the $$A.P.:b_{1},b_{2},....,b_{n}$$. If $$b_{31}=-277,b_{43}=-385 \text{ and } a_{78}=327$$ then $$a_{1}$$ is equal to

Let, $$d_a$$ and $$d_b$$ be the common differences of the two APs.

We have,

$$ b_{43}-b_{31} = (b_1+42d_b) - (b_1+30d_b) = 12d_b $$

$$ \Rightarrow 12d_b =-385 - (-277) = -108 \Rightarrow d_b = -9 $$

And, $$ d_a = d_b + 13 = -9 + 13 = 4 $$

Now, $$ a_{78} = a_1 + 77d_a = 327 $$

$$\Rightarrow a_1 + 77 \cdot 4 = 327$$

$$ \Rightarrow a_1 = 327 - 308 = 19 $$

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