U0=48 Un=0.75un-1+25 Find the long run value

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U0=48 Un=0.75un-1+25 Find the long run value

Guest Sep 23, 2014

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Best Answer

+33653

+5

U0 = 48

U1 = (3/4)U0 + 25

U2 = (3/4)U1 + 25 = (3/4)²U0 + 25(1 + 3/4)

U3 = (3/4)U2 + 25 = (3/4)³U0 + 25(1 + 3/4 + (3/4)²)

...

Un = (3/4)ⁿU0 + 25(1 + 3/4 + (3/4)² + ... + (3/4)^n-1)

As n tends to infinity (3/4)ⁿ tends to 0 and the sum in brackets tends to 1/(1 - 3/4) = 4, so Un tends to 25*4 = 100

Alan Sep 24, 2014

1⁺⁰ Answers

+33653

+5

Best Answer

U0 = 48

U1 = (3/4)U0 + 25

U2 = (3/4)U1 + 25 = (3/4)²U0 + 25(1 + 3/4)

U3 = (3/4)U2 + 25 = (3/4)³U0 + 25(1 + 3/4 + (3/4)²)

...

Un = (3/4)ⁿU0 + 25(1 + 3/4 + (3/4)² + ... + (3/4)^n-1)

As n tends to infinity (3/4)ⁿ tends to 0 and the sum in brackets tends to 1/(1 - 3/4) = 4, so Un tends to 25*4 = 100

Alan Sep 24, 2014

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