Question

find common difference of an AP whose first term is 5 and the sum of its first four terms is half the sum of the next four term is ????​

Answer 1

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❏ Question:-

Find common difference of an AP whose first term is 5 and the sum of its first four terms is half the sum of the next four term is ????

❏ Solution:-

✏ Given:-

• first term (a) = 5
• $S_4=\dfrac{S_{5-8}}{2}$

✏ To Find:-

• Find Common Difference (d) =?

✏ Explanation :-

• Now, sum of first 4 terms is ,

$\sf\implies S_4=\dfrac{\cancel4[2\times5+(4-1)d]}{\cancel2}$

$\sf\implies S_4=2[10+3d]$

$\sf\implies S_4=(20+6d)$

•Now, sum next four terms is,

$\sf\implies S_{5-8}=S_8-S_4=\dfrac{\cancel8[2\times5+(8-1)d]}{\cancel2}-S_4$

$\sf\implies S_{5-8}=4[10+7d]-2[10+3d]$

$\sf\implies S_{5-8}=(40+28d)-(20+6d)$

$\sf\implies S_{5-8}=40+28d-20-6d)$

$\sf\implies S_{5-8}=(20+22d)$

Now, according to the condition,

$\sf\implies S_4=\dfrac{S_{5-8}}{2}$

$\sf\implies 20+6d=\dfrac{20+22d}{2}$

$\sf\implies 20+6d=10+11d$

$\sf\implies 20-10=11d-6d$

$\sf\implies 10=5d$

$\sf\implies d=\dfrac{\cancel{10}}{\cancel5}$

$\sf\implies \boxed{\red{\large{d=2}}}$

∴ Common Difference is = 2 (answer).

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❏ Formula Used :-

A.P. Series

If in an A.P. series "a" be the first term and "d" be the common difference then ,

(1) The n'th term is given by the formula .

$\sf\longrightarrow\boxed{ a_n=a+(n-1)d }$

(2)Sum of n number of terms ,

$\sf\longrightarrow\boxed{ S_n=\frac{n[2a+(n-1)d]}{2}}$

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Answer 2

2 is the common difference of an AP .

Given:

a (first term of the arithmetic progression) = 5

$S_{4}=\frac{1}{2}(S_{8}-S_{4})$

To find:

d (Common Difference) = ?

Solutions:

The general sequence of an AP is a ,a + d ,a + 2d ,a + 3d,…

Substituting a=5 then

5, 5 + d,5 + 2d,5 + 3d,5 + 4d,5 + 5d,5 + 6d,5 + 7d,,..

Let the first 4 terms be 5,5 + d,5 + 2d,5 + 3d

And let the next 4 terms be = 5 + 4d,5 + 5d,5 + 6d,5 + 7d

And $\bold{S_{4}=\frac{1}{2}(S_{8}-S_{4})}$----(1)

By substituting these values in (1)

$\begin{array}{l}{5+5+d+5+2 d+5+3 d} \\ {\qquad \qquad=\frac{1}{2}(5+4 d+5+5 d+5+6 d+5+7 d)}\end{array}$

20+6d=10+11d

10=5d

d=2

Therefore, the common difference = 2

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