Question

(ii) Find the common difference of an AP whose first term is 5
and the sum of the first four terms is half the sum of the next
four terms.
​

Answer 1

Answer:

Answer:

Given:-

First term = 5 = a

Sum of first four term = 1/2sum of next four term

Solution:-

Let d is common difference of AP

Now first 4term is 5, 5+d, 5+2d, 5+3d

and next 4term 5+4d, 5+5d, 5+6d, 5+7d

Now according to question

20 + 6d = (20 + 22d)/2

20 + 6d = 10 + 11d

d = 2

Answer 2

Explanation:

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})S

4

=

2

1

(S

8

−S

4

)

To find:

d (Common Difference) = ?

Solution:

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})}S

4

=

2

1

(S

8

−S

4

) ----(1)

By substituting these values in (1)

\begin{gathered}\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}\end{gathered}

5+5+d+5+2d+5+3d

=

2

1

(5+4d+5+5d+5+6d+5+7d)

20+6d=10+11d

10=5d

d=2

Therefore, the common difference = 2

\begin{gathered}\\\\\\\end{gathered}

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