Question

Help me please !!

The sum of 5th and 9th terms of an A.P. is 72 and the sum of 7th and 12th terms is 97. Find the A.P.

Answer 1

`\star\:{\underline{\underline{\sf{\purple{Given ::}}}}}`

❖
`\sf a_(5) + a_(9) = 72`

❖
`\sf a_(7) + a_(12)= 97`

`\star\:{\underline{\underline{\sf{\purple{To \: Find ::}}}}}`

❖ The A.P

`\star\:{\underline{\underline{\sf{\purple{Solution ::}}}}}`

Let,

`a` be the first term and
`d` be the common difference of the A.P

According to the question,

`\sf a_(5) + a_(9) = 72` and
`\sf a_(7) + a_(12)= 97`

`\longrightarrow \sf (a + 4d) + (a + 8d) = 72` and
`\sf (a + 6d) + (a + 11d) = 97`

Thus, we have

`\longrightarrow \sf 2a + 12d = 72 - - (i)`

`\longrightarrow \sf 2a + 17d = 97 - - (ii)`

Subtracting (i) from (ii), we get

`\implies \sf 5d = 25`

`\implies \sf d = (25)/(5)`

`\implies {\star{ \underline{\boxed{\sf{\pink{\sf d = 5}}}}}}`

Now,

Putting d=5 in (i), we get

`\longrightarrow \sf 2a + 12(5) = 72`

`\longrightarrow \sf 2a + 60= 72`

`\longrightarrow \sf 2a= 72 - 60`

`\longrightarrow \sf 2a= 12`

`\longrightarrow \sf a= (12)/(2)`

`\longrightarrow{\star{ \underline{\boxed{\sf{\pink{\sf a = 6}}}}}}`

`\therefore` a=6 and d=5

Hence, the A.P is 6,11,16,21,26...

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