Question

2²+4²+6²+.......+(2n)²​

Answer 1

Answer:

2 n ( n +1 ) ( 2n + 1 ) / 3

Step-by-step explanation:

To find---> Value of

2² + 4² + 6² +..............+ (2n)²

Solution--->

ATQ,

2² + 4² + 6² + ..............................+ ( 2n )²

= ( 2 × 1 )² + ( 2 × 2 )² + ( 2 × 3 )² +............. +( 2 × n )²

= 2² × 1² + 2² × 2² + 2² × 3² +...............+2² × n²

= 2² ( 1² + 2² + 3² +................................ + n² )

We have a formula

1² + 2² + 3² +...........+n² =n (n+1) (2n+1) / 6

Using it we get,

= 4 { n ( n + 1 ) ( 2n + 1 ) / 6 }

= 4 n ( n + 1 ) ( 2n + 1 ) / 6

= 2 n ( n + 1 ) ( 2n + 1 ) / 3

Additional formulee--->

1) 1+2+3........+n = n ( n + 1 ) / 2

2) 1³+2³+3³+........+n³ = n² ( n + 1 )² / 4

3) Sum of n terms of AP

Sₙ = n/2 {2a + ( n - 1 ) d }

4) Sum of n terms of GP

Sₙ = a ( rⁿ - 1 ) / ( r - 1 )

(5) Sum of infinite terms of GP

S(infinite terms ) = a / ( 1 - r )

Answer 2

Answer:

Step-by-step explanation:

2² + 4² + 6² + ..............................+ ( 2n )²

= ( 2 × 1 )² + ( 2 × 2 )² + ( 2 × 3 )² +............. +( 2 × n )²

= 2² × 1² + 2² × 2² + 2² × 3² +...............+2² × n²

= 2² ( 1² + 2² + 3² +................................ + n² )

By using a  formula

1² + 2² + 3² +...........+n² =n (n+1) (2n+1) / 6

Using it we get,

= 4 { n ( n + 1 ) ( 2n + 1 ) / 6 }

= 4 n ( n + 1 ) ( 2n + 1 ) / 6

= 2 n ( n + 1 ) ( 2n + 1 ) / 3