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grandymaker [24]
2 years ago
9

*Find the formula of the series 1+2²+3²+4²+....+n²*​

Mathematics
1 answer:
Sophie [7]2 years ago
3 0

The formula is \frac{n(n+1)(2n+1)}{6}

What are series?

In mathematics, we can describe a series as adding infinitely many numbers or quantities to a given starting number or amount.

We will find the formula as shown as below:

Let S=1+2^2+3^2+4^2+................+n^2

We know (n+1)^3=n^3+3n^2+3n+1

(1+1)^3=1^3+3(1)^2+3(1)+1

(2+1)^3=2^3+3(2)^2+3(2)+1

(3+1)^3=3^3+3(3)^2+3(3)+1

.

.

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

On adding

2^3+3^3+4^3......(n+1)^3=(1^3+2^3+3^3+.....+n^3)+3(1^2+2^2+.....+n^2)+3(1+2+3....n)+(1+1+1+....+1)

2^3+3^3+4^3......(n+1)^3-(1^3+2^3+3^3+.....+n^3)=3S+\frac{3n(n+1)}{2}+(1+1+1+....+1)

(n+1)^3-1^3=3S+\frac{3n(n+1)}{2} +n

n^3+3(n)^2+3(n)+1-1=3S+\frac{3n(n+1)}{2} +n

2n^3+6n^2+6n=6S+3n^2+3n+2n

6S=2n^3+3n^2+n

6S=2n^2(n+1)+n(n+1)

6S=(n+1)(2n^2+n)

6S=n(n+1)(2n+1)

S=\frac{n(n+1)(2n+1)}{6}

Hence, the formula is \frac{n(n+1)(2n+1)}{6}

Learn more about Series here:

brainly.com/question/24643676

#SPJ1

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