Calculate Sn (x)Calculate Sn (x) = x + 2x^2 + 3x^3 +...+nx^n if x is real and n is natural.

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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We'll calculate Sn (x) for x = 1.

Sn (x) = 1 + 2*1 + 3*1 + .... + n*1

Since the expression above represents the sum of the first natural n terms, we'll replace it by the formula:

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

Now, we'll multiply Sn (x) by x:

xSn (x) = x^2 + 2x^3 + 3x^4 + ... + (n-1)x^n + nx^(n+1)

We'll compute the difference:

Sn (x) - xSn (x) = x + x^2 + x^3 + ...+x^n - nx^(n+1)

We'll recognize the sum of the terms of the geometric progression, whose common ratio is x:

x + x^2 + x^3 + ...+x^n = [x - x^(n+1)]/(1-x)

Sn (x) - xSn (x) = [x - x^(n+1)]/(1-x) - nx^(n+1)

Sn (x) - xSn (x) = [x - (n+1)x^(n+1) + nx^(n+2)]/(1-x)

If x = 1 => Sn(x) = n(n+1)/2

For any value of x, except 1:

Sn(x) = [nx^(n+2) - (n+1)x^(n+1) + x]/(1-x)

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