Question

Without adding, find the sum
(i) 1+3+5+7+9
(ii) 1+3+5+7+9+11+13+15+17+19
(iii) 1+3+5+7+9+11+13+15+17+19+21+23

Answer 1

Hint: In this question we have given some series of odd numbers and we have to find their sum without adding, so for finding the sum of first n odd natural number without actually adding, so fo that we have to know one method that is, if you have given a series where we take the summation of first n odd natural numbers i.e, (1+3+5+7+9+ . . . . . +nth odd number),
Then for this first of all you have to count that how many odd number you have given to sum up, so from the above series we can easily say that the total number of odd number are n, so we can write,
(1+3+5+7+9+ . . . . . +nth odd number) = $n^{2}$ ……..equation(1)

Complete step-by-step solution:
Firstly we are going to solve for question(i),
(i) So the given series is, 1+3+5+7+9
So in the above series ,first of all we have to count how many odd natural number we have given,
So,
first term=1
Second term=3
Third term=5
Fourth term=7
Fifth term=9
So we can say that in the above series we have given in total 5 terms, or in other wards we can say that we have to find the sum of first 5 odd natural number, so by equation(1) we can say that n=5,

Therefore, 1+3+5+7+9= $5^{2}$ = $5\times 5$ = 25.
So, 25 is our required solution for the first series.

(ii) Given series, 1+3+5+7+9+11+13+15+17+19
So in similar way we have to count how many terms are there, so after counting we can easily say that the total odd numbers are 10, i.e, n=10,
So by equation (1) we can write,
1+3+5+7+9+11+13+15+17+19 = $10^{2}$ = $10\times 10$ =100.
So, 100 is the solution for the question number (ii).

(iii) Given series, 1+3+5+7+9+11+13+15+17+19+21+23,
In this series, by counting we can find that the total odd numbers are 12, i.e, n=12,
So by equation (1) we can write,
1+3+5+7+9+11+13+15+17+19+21+23 = $12^{2}$ = $12\times 12$ =144.
So, 144 is the solution for the question number (iii).

Note: To solve this type of question you have to first identify whether the odd numbers are in chronological order or not, if all the odd numbers are present without any missing then only we can apply the formula which we have mentioned in equation(1).