Question

please answer the question​

Answer 1

Question :

Sum of 1 to n natural numbers is 45, then find the value of n.

Answer :

The required value of n is 9 ( option [a] 9 )

Step-by-step explanation :

Given :

Sum of 1 to n natural numbers is 45

To find :

the value of n

Solution :

Natural numbers includes all the positive integers from 1 to infinity.

The sequence of the natural numbers is  $\bf 1,2,3,...,n$

We observe that each term is obtained by adding 1 to the previous term. Hence, the sequence is in Arithmetic Progression. (A.P)

 

In an A.P., sum of n terms is given by,

$\underline{\boxed{ \bf S_n=\dfrac{n}{2}[2a+(n-1)d]}}$

where

a denotes first term

d denotes common difference ( i.e., difference between a term and it's preceding term)

In the sequence 1 , 2 , 3 ,... , n

first term, a = 1

common difference, d = 2 - 1 = 3

Substitute the values in the formula,

$\tt S_n=\dfrac{n}{2}[2(1)+(n-1)(1)] \\\\ \tt 45=\dfrac{n}{2}[2+n-1] \\\\ \tt 45=\dfrac{n}{2}[1+n] \\\\ \tt 45 \times 2=n[1+n] \\\\ \tt 90=n+n^2 \\\\ \tt n^2+n-90=0$

Solving the quadratic equation,

n² + n - 90 = 0

n² + 10n - 9n - 90 = 0

n(n + 10) - 9(n + 10) = 0

(n + 10) (n - 9) = 0

⇒ n + 10 = 0 ; n = -10

⇒ n - 9 = 0 ; n = +9

Since n is a natural number, n must be positive.

Hence, the value of n is 9