Question

We can express the number 1681 as the sum of first ...... odd natural number

Answer 1

We can express the number 1681 as the sum of first 41 odd natural numbers

Step-by-step explanation:

We know that odd natural numbers are

$1,3,5,7,9,.....$

Which constitutes an AP

The first term of this AP is 1 and common difference is 3 - 1 = 2

Let the sum of first n odd natural numbers be 1681

We know that sum of n terms of an AP whose first term is a and common difference is d is given by

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

Thus,

$1681=\frac{n}{2}[2\times 1+(n-1)\times 2]$

$\implies 1681=n(1+n-1)$

$\implies n^2=1681$

$\implies n^2-41^2=0$

$\implies n=-41, 41$

But n cannot be negative

Therefore, n = 41

Hope this answer is helpful.

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Answer 2

Sum of  41 terms of the odd natural numbers is 1681.

Step-by-step explanation:

Given: We can express the number 1681 as the sum of first ...... odd natural number.

Find: Find n.

Solution:

The AP series is as follows: We know that odd natural numbers are

1, 3, 5, 7, 9, .....

We find that a = first term = 1

d = common difference = 2

Sum of n odd numbers = 1681

We know that the formula for Sum of AP = n/2 [2a + (n-1)d]

1681 = n/2 [2 + (n-1)2]

 1681 * 2 = n [ 2 + 2n - 2]

 1681 * 2 = 2n²

 1681 = n²

Therefore n = root of 1681 = root of 41² = 41

Sum of  41 terms of the odd natural numbers is 1681.