When $9^{t h}$ term of an A.P. is divided by its $2^{n d}$ term the quotient is 5 & when $13^{t h}$ term is divided by the $6^{t h}$ term, the quotient is 2 and remainder is 5. The sum of first term and the common difference of the A.P. will be equal to .......... ?

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Solution

let the first term of AP is $a$ and the common difference is $d$

$9 t h$ term $= a + 8 d$

$2 n d$ term $= a + d$

$\frac{a + 8 d}{a + d} = 5$

$a + 8 d = 5 a + 5 d$

$4 a = 3 d$ ....(1)

$13 t h$ term $= a + 12 d$

$6 t h$ term $= a + 5 d$

$a + 12 d = 2 \times (a + 5 d) + 5$

$0.75 d + 12 d = 2 \times (0.75 d + 5 d) + 5$ (from equation (1), putting value of $a$ in terms of $d$ )

$12.75 d = 11.5 d + 5$

$1.25 d = 5$

$d = 4$

from equation (1)

$a = 3$

sum of first term and the common difference is $a + d = 3 + 4 = 7$

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When 9th term of an A.P. is divided by its 2nd term the quotient is 5 & when 13th term is divided by the 6th term, the quotient is 2 and remainder is 5. The sum of first term and the common difference of the A.P. will be equal to .......... ?

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