Question

The sum of a certain number of consecutive odd num- bers, starting with 1, is 5184. How many odd numbers are added?

Answer 1

72 consecutive odd numbers are added.

Explanation

The consecutive odd numbers starting with 1 are........

$1, 3, 5, 7, 9, 11, ..........................$

We can see the above series is in Arithmetic series, with common difference$(d)$ $= 3-1=5-3= 2$

Suppose, the sum of $n$ terms in that series is 5184.

Formula for Sum of  $n$  terms in arithmetic series is:  $S_{n}= \frac{n}{2}[2a_{1}+(n-1)d]$

Here,  $S_{n}= 5184, a_{1}= 1$ and $d= 2$

So, plugging these values into the above formula......

$5184= \frac{n}{2}[2(1)+(n-1)(2)]\\ \\ 5184= \frac{n}{2}[2+2n-2]\\ \\ 5184= \frac{n}{2}(2n) \\ \\ 5184=n^2\\ \\ n=\sqrt{5184}=72$

Thus, 72 consecutive odd numbers are added.

Answer 2

Given :  sum of some consecutive odd number starting with 1,is 5184.

To Find : how many odd numbers are added

Solution:

sum of n consecutive odd number starting with 1 is n²

=> n²  = 5184

=> n = 72

Proof

Let say n numbers

a = 1  , d = 2

nth term = a + (n - 1)d

last number = 1  + (n - 1) 2  = 2n  - 1

Sum = (n/2)(1 + 2n - 1)

= n²

72 odd numbers are added

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