Question

The 8th term of an ap is 5 times the 4th term.the sum of first 20 terms is 450.find the sun of first 25 term?

Answer 1

The sum of the first 25 terms is `750`

Explanation:

Recall that each consecutive term in an arithmetic sequence is the previous term plus a constant `d`.

Therefore,

`a_2 = a_1 + d`
`a_3 = a_1 + 2d`

etc.

Therefore:

`a_4 + 4d = 5a_4`

Simplifying we get

`4d = 4a_4`

`d = a_4`

Now recall that the sum of the first `n` terms of an ap is given by `s_n = n/2(2a_1 + (n - 1)d)`. However, we have one more variable than we can deal with with a system of two equations. We can rewrite `a_1` as `a_4 - 3d`.

`450 = 20/2(2(a_4 - 3d) + 19(a_4))`

`450 = 10(2(a_4 - 3a_4) + 19a_4)`

`45 = 2a_4 - 6a_4 + 19a_4`

`45 = 15a_4`

`3 = a_4`

Since `a_1 = a_4 - 3d`, and `d = a_4` we get `a_1 = 3 - 3(3) = -6`.

Now we have everything we need to find the sum of the first 25 terms.

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

`s_25 = 25/2(2(-6) + (25 - 1)3)`

`s_25 = 25/2(60)`

`s_25 = 750`

Hopefully this helps!