Question

without doing actual addition find the sums of 1+3+5+7+9+.......+39+41

Answer 1

Use formula of sum of A.P.

S = n/2 (a+l)

where a is 1st element, n is total no of elements, and l is last element.
So,

A = 21/2(41+1)
A = 21*21
A = 441

Answer 2

The sum of 1 + 3 + 5 + 7 + 9 +.......+ 39 + 41 without doing actual addition is 441.

Concept used:

The square of a natural number 'n' is equal to the sum of the first n odd numbers.

For example :

1² = 1

2² = 1 + 3, sum of first 2 odd numbers

3² = 1 + 3 + 5, sum of first 3 odd numbers

4² = 1 + 3 + 5 + 7,  sum of first 4 odd numbers

5² = 1 + 3 + 5 + 7 + 9 , sum of first 5 odd numbers

Formula used:

In general, for any natural number n

n² = sum of the first n odd numbers

Given:

1 + 3 + 5 + 7 + 9 +.......+ 39 + 41

To find :

Find the sum of given numbers without doing actual addition.

Solution:

Step 1: Write the odd numbers from 1 to 50:

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41,43,47,49.

Step 2: Count the odd numbers till 41 from the above odd numbers and find the sum :

Here, n = 21

The sum of 1 + 3 + 5 + 7 + 9 +.......+ 39 + 41

n² = sum of the first n odd numbers

21² = 1 + 3 + 5 + 7 + 9 +.......+ 39 + 41, sum of the first 21 odd numbers

441 = 1 + 3 + 5 + 7 + 9 +.......+ 39 + 41

Hence, the sum of 1 + 3 + 5 + 7 + 9 +.......+ 39 + 41 is 441.

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