Question

In an AP: Given a = 5, d = 3, a_n = 50, find n and S_n.

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

n and S_n, if a = 5, d = 3 and a_n = 50.

Answer 1

Given that, a = 5, d = 3, an = 50

As a_n = a + (n − 1)d,

⇒ 50 = 5 + (n - 1) × 3

⇒ 3(n - 1) = 45 

⇒ n - 1 = 15

⇒ n = 16

Now, S_n = `n/2 (a + a_n)`

S_n = `16/2 (5 + 50)`

S_n = 440

Answer 2

 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 is 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,

S₁₆ = `(16/2) [5 + 50]`

= 8(55)

= 440

Therefore, for the given A.P n = 16 and S₁₆ = 440