Question

the sum of 5th and 9th terms of an AP is 72 and the sum of 7th and 12th terms is 97. find the AP.

Answer 1

Given that ,

● The sum of 5th and 9th terms of an A.P = 72

● The sum of 7th and 12th terms of an A.P. = 97.

Find A.P. = ?

According to the question ,

♧ In the first case ,

$\textbf{An = a + ( n - 1 ) d}$

The sum of 5th and 9th term =>

[ a + ( 5 - 1 ) d ] + [ a + ( 9 - 1 ) d ] = 72

[ a + 4d ] + [ a + 8d ] = 72

[ 2a + 12d ] = 72 ...................( 1 )

♧ In second case ,

The sum of 7th and 12th term = 97

[ a + ( 7 - 1 ) d ] + [ a + ( 12 - 1 ) d = 97

[ a + 6d ] + [ a + 11d ] = 97

[ 2a + 17d ] = 97 ................( 2 )

Now ,

On subtracting eq. ( 2 ) from ( 1 ) ,

we get ,

2a + 17d = 97
2a + 12d = 72
____________
_×_ 5d = 25

or , d = 5

□ $\textbf{hence , difference = 5}$

Put the value of " d " on eq. ( 2 )

we get ,

2a + 12d = 72

2a + 12 ( 5 ) = 72

2a + 60 = 72

2a = 72 - 60

2a = 12

a = 12 / 2

or , a = 6

□ $\textbf{hence , the first term is 6}$.

So the A.P. will be =>

$\textbf{a + a + d + a + 2d + a + 3d + a + 4d and so on.}$

■ $\textbf{6 , 11 , 16 , 21 , 26 , 31 , 36 .....}$


$\textbf{Thanks !!!!}$