The common difference of the A.P.: a_{1}, a_{ | vedclass

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(C) Let the common differences of the two $A.P.$s be $d_{1}$ and $d_{2}$ respectively.Given that $d_{1} = d_{2} + 13$.For the $A.P.$ $b_{n}$, we have

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$19$

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(C) Let the common differences of the two $A.P.$s be $d_{1}$ and $d_{2}$ respectively.Given that $d_{1} = d_{2} + 13$.For the $A.P.$ $b_{n}$, we have $b_{31} = b_{1} + 30d_{2} = -277$ (Equation $1$) and $b_{43} = b_{1} + 42d_{2} = -385$ (Equation $2$).Subtracting Equation $1$ from Equation $2$:$(b_{1} + 42d_{2}) - (b_{1} + 30d_{2}) = -385 - (-277)$$12d_{2} = -108$$d_{2} = -9$.Thus, $d_{1} = -9 + 13 = 4$.For the $A.P.$ $a_{m}$, we have $a_{78} = a_{1} + 77d_{1} = 327$.Substituting $d_{1} = 4$:$a_{1} + 77(4) = 327$$a_{1} + 308 = 327$$a_{1} = 327 - 308 = 19$.

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 $b_{31} = -277$, $b_{43} = -385$ and $a_{78} = 327$, then $a_{1}$ is equal to

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