Let there be an A.P. with the first term ‘a’, common difference’. If a denotes its nth term and Sn the sum of first n terms, find

n and S_{n}, if a = 5, d = 3 and a_{n} = 50.

#### Solution

Here, we have an A.P. whose n^{th} term (a_{n}), first term (a) and common difference (d) are given. We need to find the number of terms (n) and the sum of first n terms (S_{n}).

Here,

First term (*a*) = 5

Last term (`a_n`) = 50

Common difference (*d*) = 3

So here we will find the value of *n* using the formula, `a_n = a + (n -1)d`

So, substituting the values in the above-mentioned formula

50 = 5 + (n -1)3

50 = 5 = 3n - 3

50 = 2 + 3n

3n = 50 - 2

Further simplifying for n

3n = 48

`n = 48/3`

n = 16

Now, here we can find the sum of the *n* terms of the given A.P., using the formula,

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

Where *a* = the first term

*l* = the last term

So, for the given A.P, on substituting the values in the formula for the sum of n terms of an A.P., we get,

`S_16 = (16/2)[5 + 50]`

= 8(55)

= 440

Therefore, for the given A.P n = 16 and `S_16 = 440`