Obtain the sum of the first 56 terms of an A.P. Whose 18th and 39th terms are 52 and 148 respectively.
(1) Usint `t_(18)` and `t_(39)` find two simultaneous equations in variables a and d.
(2) Using these equations, find `S_(56)`

Answer

The sum of the first 56 terms is 5600

Explanation

Let the first term of A.P. be a and the common difference d.
`t_(n) = a+ ( n -1) d ` …(Formula)
`:. T_(18) = a + ( 18-1) d `
`:. 52 = a+ 17d` ....(1)
and `t_(39) a + ( 39-1)d`
`:. 148 = a + 38d` ...(2)
Adding equations (1) and (2)
`52= a + 17d ` ...(1)
`148 = a + 38 d ` ...(2)
`bar( 200 = 2a + 55d)` ....(3)
We have to find `S_(56 )`
`S_(n) = ( n )/( 2) [2a + (n -1) d]`
`:. S_(56) = ( 56)/( 2) [2a + ( 56-1) d ]`
`= 28 (2a+ 55d]`
`= 28( 200) ` ...[From (30]
` = 5600`