Question

What is the common difference? if an A. P has the sum of first and last term is 43 and the third is 11 the sum is 258

Answer 1

The first term is `35/18`

The common difference is `32/9`

Explanation:

Suppose we denote the `n^(th)` term of the AP by `u_n` where `u_n=a+nd`, making the AP sequence:

`{a, a+d, a+2d, a+3d, ..., a+(n-1)d }`

Given the sum of the first and last term is `43`, we have:

`(a) + (a+(n-1)d) = 43`

`:. 2a+(n-1)d = 43` ..... [A]

Given, also, that the third term is `11`, we have:

`a + 2d = 11` ..... [B]

Given, also, that the sum is `258`, and utilising the AP summation formula:

`S_n = n/2(2a+(n-1)d)`

We have:

`n/2(2a+(n-1)d) = 258`

`n(2a+(n-1)d) = 516` ..... [C]

Substituting [A] into [C] we get:

`n(43) = 516 => n =12`

Back substitution of `n=12` into [A] we get:

`:. 2a+11d = 43` ..... [D]

So that [D]-2[B] yields:

`11d-2d = 43-11 => 9d = 32`

`:. d = 32/9`

Finally, substituting `d=32/9` into [D]:

`2a+(11*32)/9 = 43 => 2a = 43 - 352/9`

`:. a = 35/18`