Question

The sum of 3rd and 5th term of an ap is 16 and their product is 60.find the terms of the ap?

Answer 1

`t_3 = 6`, `t_5 = 10`
OR
`t_3 = 10`, `t_5 = 6`

Explanation:

In an arithmetic progression, we add a common difference to each term to get the next term. Therefore,

`t_3 + 2d = t_5`

Letting `t_3 = x`, we see that

`x+ 2d = t_5`

Now we can write down a system of equations

`{(x+ 2d + x = 16), (x(x + 2d) = 60):}`

We simplify the first equation to `2x + 2d = 16 -> x + d = 8`

This means that `d = 8 - x`. We substitute into the second equation to get

`x(x + 2(8 - x)) = 60`

`x(x + 16 - 2x) = 60`

`x(16 - x) = 60`

`16x - x^2 = 60`

`0 = x^2 - 16x + 60`

`0 = (x- 10)(x - 6)`

`x =10 or 6`

We have a couple of different answers here. If `x= 10`, than `d = 8 - 10 = -2` . Alternatively, if `x = 6` then `d = 8 -6 = 2`.

The terms of the arithmetic progression are therefore:

`6, 10`

It works out to be the same thing if `d = +2` or `d = -2`.

Hopefully this helps!