Question

The sum of n natural numbers is 325. Find n

Answer 1

Answer:

n = 25

Step-by-step explanation:

$\frac{n(n+1)}{2} = 325\\n^{2}+n=650\\n^2+n-650=0\\n^2+n-(25*26)=0\\n^2-25n+26n-(25*26)=0\\n(n-25)+26(n-25)=0\\(n-25)(n+26)=0\\n-25=0 \hspace{5} (or)\hspace{5} n+26=0\\n=25\hspace{5} or\hspace{5} n=-26\\n\hspace{5} is\hspace{5} natural \hspace{5}number.\\so\hspace{5}n=25$

Answer 2

Sum of 25 natural numbers is 325

Given :  sum of n natural numbers is 325.

To Find : Value of n

Solution :

natural numbers are counting numbers 1 ,2  , 3

Arithmetic sequence

Sequence of terms in which difference between one term and the next is a constant.

This is also called Arithmetic Progression AP

Arithmetic sequence can be represented in the form :

a, a + d  , a + 2d , …………………………, a + (n-1)d

a = First term

d = common difference = aₙ-aₙ₋₁

nth term =  aₙ =  a + (n-1)d  

Sₙ = (n/2)(2a + (n - 1)d)

Sum of n  natural numbers  

(n/2)(2 * 1 + (n - 1) * 1)

= (n/2)(2 + n - 1)

= (n )(n + 1)/2

n(n + 1)/2  = 325

=> n² + n - 650 = 0

=> n² + 26n - 25n -  650  =0

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

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

=> n = 25 , n = - 26

n can not be negative

Hence Value of n is 25

Sum of 25 natural numbers is 325

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