help with this math question

Question

A series has sum Sn =n(2n-1)

i) find, in terms of n, the nth term of the series.

ii) Hence, show that the series is an AP

 

c. Find an expression for the sum to infinity of the Geometric Progression.

1+2x+4x2+8x3+....

 

Last one.

A sequence {Un} is generated by the recurrence relation Un+1= 8÷U_n - 3 where n≥1

 a) given that U₂ = 2U₁ Find values for U₁ 

Answer 1

For the series S_n =n(2n-1), we have

 

i) a_n = S_n - S_n-1 = n(2n-1) - (n - 1)[2(n - 1) - 1]

 

      = n(2n-1) - (n - 1)(2n - 3)

 

      = (2n² - n) - (2n² - 5n + 3)

 

      = 4n - 3

 

ii) The terms in the series indeed form an arithmetic progression, with first term 1, and an increment of 4.

 

    a₁ = 1

 

    a_n+1 - a_n = [4(n+1) - 3] - (4n - 3) = 4

 

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Geometric series:

 

           S = 1 + 2x + 4x² + 8x³ + ...

        2xS =       2x + 4x² + 8x³ + ...

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(1 - 2x)S = 1

 

S = 1/(1 - 2x)

 

Also, this sum converges to the above as long as |x| < 1/2. Otherwise, it diverges.

 

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Recursive Sequence: U_n+1 = 8/(U_n - 3) and U₂ = 2U₁

 

I am assuming you meant to put parentheses on the right hand side.

 

Plug in the constraint into the recursion.

 

2U₁ = 8/(U₁ - 3)

 

U₁(U₁ - 3) - 4 = 0

 

U₁² - 3U₁ - 4 = 0

 

(U₁ + 1)(U₁ - 4) = 0

 

U₁ = -1 or 4.