Question

4
10) If S, S, and S, are the sums of first n natural
numbers, their squares and their cubes
respectively then show that -
95,2 = S, (1+8 S.).
5



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Answer 1

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Step-by-step explanation:

We have formulas to calculate the sum of first n natural numbers , their squares and also their cubes.

So, S

1

= Sum of first n natural numbers =

2

n(n+1)

S

2

= Sum of squares of first n natural numbers =

6

n(n+1)(2n+1)

S

3

= Sum of cubes of first n natural numbers =(

2

n(n+1)

)

2

So,

S

2

2

S

3

(1+8S

1

)

=

(

6

n(n+1)(2n+1)

)

2

(

2

n(n+1)

)

2

(1+8(

2

n(n+1)

))

On simplifying, we get

S

2

2

S

3

(1+8S

1

)

=

(2n+1)

2

9(4n

2

+4n+1)

=

(2n+1)

2

9(2n+1)

2

=9