The common difference of the A.P.: $a_{1},a_{2},\ldots,a_{m}$ is 13 more than the common difference

Question

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

Accepted Answer

Let common difference of A.P.'s are $d_{1}\&{\text{ }d}_{2}$

$$\therefore d_{1} = 13 + d_{2}$$

$$\begin{array}{r} b_{1} + 30{\text{ }d}_{2} = - 277\#(1) \end{array}$$

$$\begin{array}{r} b_{1} + 42{\text{ }d}_{2} = - 385\#(2) \end{array}$$

By (2) - (1)

$${12{\text{ }d}_{2} = - 108 }{d_{2} = - 9 }{\therefore d_{1} = 4 }$$Now $a_{78} = 327$

$${\Rightarrow a_{1} + 77{\text{ }d}_{1} = 327 }{\Rightarrow a_{1} + 308 = 327 }{a_{1} = 19}$$

Question

Options

Solution

More Progression (Sequence and Series) Questions