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Sequences and Series

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Question 67

A company’s net cash flow in year nnn, denoted by PnP_nPn​ (in millions of dollars), follows a geometric progression with common ratio rrr.

You are given that:

  • The sum of the cash flows in the second and third years is P2+P3=10P_2 + P_3 = 10P2​+P3​=10
  • The cash flow in the fourth year is P4=20P_4 = 20P4​=20
a.

Show that r r\,r satisfies the equation r2−2r−2=0r^2 - 2r - 2 = 0r2−2r−2=0.

[4]
b.

Given that the sum of the cash flows over an infinite time period, ∑n=1∞Pn\sum_{n=1}^{\infty} P_n∑n=1∞​Pn​, is convergent,

find the exact value of P1P_1P1​.

[4]
c.

Hence, calculate the exact value of S∞S_\inftyS∞​.

[3]

Sequences and Series Questions

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