Question

7square as the sum of concecutive odd numbers

Answer 1

Answer:

$\boxed{\sf 7^2 = 1 + 3 + 5 + 7 + 9 + 11 + 13 }$

Step-by-step explanation:

We need to express 7² as the sum of consecutive odd numbers. We will try to develop a formula for the sum of first n consecutive odd numbers.

Let us take the first odd number be a , then the second would be a + 2 . Hence , the , Series will be ,

$\sf\dashrightarrow S = a , a + 2 , a + 4 , \cdots n \ terms$

If we take this series as a AP , then the common difference would be 2 and the first number a . Since the series is of first n odd numbers therefore the first number a = 1 . On using the formula to find sum of n terms of AP ,

$\sf\dashrightarrow S =\dfrac{n}{2}[2a + (n-1)d] \\\\\sf\dashrightarrow S = \dfrac{n}{2}[ 2(1) + ( n-1)2 ] \\\\\sf\dashrightarrow S =\dfrac{n}{2} [ 2 + 2n - 2 ] \\\\\sf\dashrightarrow S = \dfrac{n}{2} [ 2n ] \\\\\sf\dashrightarrow \boxed{\red{\sf S = n^2}}$

Therefore the sum of first n odd numbers is n² . In a similar way here the the sum is 7². So this will be the sum of first 7 odd numbers .

$\rule{200}7$

$\qquad\qquad\tiny\sf\red{\dashrightarrow} \:\: \boxed{\boxed{\sf 7^2 = \pink{ 1 + 3 + 5 + 7 + 9 + 11 + 13 } }}$

$\rule{200}7$