How many terms of the series ` 54 + 51 + 48 + …` must be taken to make 513 ? Explain ? Explain the -double anwer.

Explanation

Let the sum to n terms of the given series be 513
Here, a = 54 , d = - 3 `S_(n) = 513 `
`because S_(n) = (n)/(2) [2a + (n - 1)d]`
`therefore 513 = (n)/(2) [2a xx54 +(n-1) xx(-3)]`
`rArr 513 = (n)/(2) [108 + 3 - 3n]`
`rArr 1026 = n [111 - 3n]`
`rArr 3n^(2)- 111n + 1026 = 0`
`rArr n^(2) - 37 n + 342 = 0`
`rArr (n - 18 ) (n - 19 ) = 0 `
` therefore n = 18 , 19 `
Explanation of double answer
` 18 ^(th)` term `= a_(18) = 54 + (18 - 1 ) (-3) = 3 `
`19 ^(th) term = a_(19) = 54 + (19 - 1) (-3) = 0`
Since ` 19^(th)` term is zero, the sum of 18 terms and the sum of 19 terms are equal.