Question

Sum of the first 'n' consecutive natural numbers is 325 how many natural numbers to be added *

Answer 1

Answer:

25.

Step-by-step explanation:

Natural numbers are: 1, 2, 3, 4, ......

This forms an A.P. as follows:

1, 2, 3, 4, .....

Here,

First Term, (a) = 1.

Common difference, (d) = 2-1 = 1.

Given: Sum of 'n' terms = 325.

Equation for relation between First term (a), Common difference (d), No. of terms (n), and the Sum of 'n' terms (Sn):

Sn = (n/2) × [2a + (n-1)d ].

Substituting the given values, we get,

325 = (n/2) × [2(1) + (n-1)(1) ] = (n/2) × [2 + n - 1 ] = (n/2) × [ 1 + n ] = (n + n²)/2.

So,

650 = n + n².

n² + n - 650 = 0.

Solving the quadratic equation,

n² + 26n - 25n - 650 = 0.

n(n+26) - 25(n+26) = 0.

(n+26)(n-25) = 0.

(n+26) = 0, or, (n-25) = 0.

n = -26, or, n = 25.

Here,

n = -26 value is not considered, because, the value of 'n' can never be negative. (since 'n' represents the number of terms, and that quantity can never take a negative value).

So,

n = 25 is taken as the answer.

Therefore,

First 25 consecutive natural numbers are to be added to obtain a sum of 325.