In an A.P. `19^(th)` term is 52 and `38^(th)` term is 128, Find sum of first 56 terms.

Answer

The sum fo the first 56 term is 5040.

Explanation

Let the first term of the A.P. be a and common difference d.
`t_(19)=52 and t_(38)=128` ….(Given)
`t_(n)=a+(n-1)d` ….(Formula)
`thereforet_(19)=a+(19-1)d`
`thereforet_(19)=a+18d=52` ….(Given, `t_(19)=52)….(1)`
and `t_(38)=a+(38-1)d`
`thereforet_(38)=a+37d=128` ....(Given, `t_(38)=128`)....(2))
adding equations (1) and (2),
`a+18d=52...(1)`
`a+37d=128` ...(2)
`2a+55d=180` ...(3)
Now, `S_(n)=(n)/(2)[2a+(n-1)d]` ...(Formula)
`thereforeS_56=(56)/(2)[2a+(56-1)d]`
`=28(2a+55d)`
`28xx180` ...[From(3)]
`thereforeS_56=5040`