Question

Prove that

1 + 5 + 9 + .....+ (4n - 3) = n(2n - 1)​

Answer 1

let

p(n) = 1 + 5 + 9 +....+ (4n - 3) = n (2n - 1)

1. If n = 1

L.H.S. 4n - 3 = 4 × 1 - 3 = 1

R.H.S. p (n) : n ( 2n - 1 ) = 1 [ 2 - 1 ] = 1 is true

:. $\bf\pink{L.H.S.~=~R.H.S.}$

2. Now we have to show p(n) is true for n + 1 also

Next term of 4n - 3 = 4 ( n + 1 ) - 3

=> 4n + 4 - 3

=> 4n + 1

Adding both sides 4n + 1

1 + 5 + 9 + ..... + ( 4n - 3 ) + ( 4n + 1 )

=> n ( 2n - 1 ) + ( 4n + 1 )

=> 2n² - n + 4n + 1

=> 2n² + 3n + 1

=> 2n² + 2n + n + 1

=> 2n ( n + 1 ) + 1 ( n + 1 )

=> ( n + 1 ) ( 2n + 1 )

=> ( n + 1 ) [ 2 ( n + 1 ) - 1 ] → 3

From 1 , 2 and 3 for all values of n

p ( n ) is true