The sum of the first 25 natural numbers is (a) 325 (b) 300 (c) 505 (d) None of these
ANSWER
Verified
Hint: To calculate the sum of the first 25 natural numbers, use the formula for calculating the sum of consecutive positive integers, which is $\sum\limits_{i=1}^{n}{i}=\dfrac{n\left( n+1 \right)}{2}$. Substitute $n=25$ in the above formula and simplify the expression to find the sum of the first 25 natural numbers.
Complete step-by-step answer:
We have to calculate the sum of the first 25 natural numbers. We know the formula for calculating the sum of consecutive positive integers, which is $\sum\limits_{i=1}^{n}{i}=\dfrac{n\left( n+1 \right)}{2}$. Substituting $n=25$ in the above formula, we have $\sum\limits_{i=1}^{25}{i}=1+2+...25=\dfrac{25\times 26}{2}$. Simplifying the above expression, we have $1+2+...+25=25\times 13=325$. Hence, the sum of the first 25 natural numbers is 325, which is option (a). We can prove the formula for calculating the sum of consecutive positive integers by applying induction on ‘n’. To do so, firstly check if the formula holds for $n=1$. Then assume that the formula holds for $n=k$ and use it to prove that the formula holds for $n=k+1$ by adding $k+1$ on both sides of the equation $1+2+...+k=\dfrac{k\left( k+1 \right)}{2}$. Simplify the equation by taking LCM to prove that the formula holds for $n=k+1$ as well.
Note: We can also solve this question by adding up the first 25 natural numbers. However, this will be very time consuming. So, it’s better to solve the question by using the formula of calculating the sum of consecutive positive integers.