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Standard XII

Mathematics

Common Difference

The sum of 4 ...

Question

The sum of 4th and 8th terms of an A.P. is 24 and the sum of the 6th and 10th terms is 34. Find the first term and the common difference of the A.P.

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Solution

Let a be the first term and d be the common difference. Then,
$a_{4} + a_{8} = 24$ [Given]
$\Rightarrow (a + 3 d) + (a + 7 d) = 24$
$\Rightarrow a + 5 d = 12$ .... (i)
$a_{6} + a_{10} = 34$
$\Rightarrow (a + 5 d) + (a + 9 d) = 34$
$\Rightarrow a + 7 d = 17$ .... (ii)
From (i) and (ii)
$a = \frac{- 1}{2}$ and $d = \frac{5}{2}$
$∴$ 1st term is $\frac{- 1}{2}$ and common difference is $\frac{5}{2}$

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The sum of 4th and 8th terms of an A.P. is 24 and the sum of the 6th and 10th terms is 34. Find the first term and the common difference of the A.P.

Let a be the first term and d be the common difference. Then, a4+a8=24 [Given] ⇒(a+3d)+(a+7d)=24 ⇒a+5d=12 .... (i) a6+a10=34 ⇒(a+5d)+(a+9d)=34 ⇒a+7d=17 .... (ii) From (i) and (ii) a=−12 and d=52 ∴ 1st term is −12 and common difference is 52