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Geometric Sequences and Series · Paper 3 — Free Maths AI HL Practice Questions

1ChallengeLAQSum of a geometric sequence10 marksPaper 3~15 min
A geometric sequence has first term u1=au_1 = a and common ratio rr, where r≠1r \neq 1. The sum of the first nn terms is SnS_n. An industrial robot is purchased for 120 000120\,000 dollars. Its value at the end of each year is modelled as a fixed percentage of its value at the start of that year, forming a geometric sequence. After 1010 years the robot is sold for 15 00015\,000 dollars.
(a)
Prove that Sn=a(1−rn)1−rS_n = \dfrac{a(1-r^n)}{1-r}. [4 marks]
(b)
Show that the value of the robot after nn complete years is 120 000⋅rn120\,000 \cdot r^n dollars, and hence calculate the annual percentage rate of depreciation, correct to three significant figures. [3 marks]
(c)
The total depreciation over the 10-year life is defined as the difference between the purchase price and the final sale value. Using your value of rr from part (b), calculate the total depreciation as a percentage of the purchase price. Evaluate whether this percentage would change if the robot had instead been purchased for 200 000200\,000 dollars but depreciated at the same annual rate for the same number of years. Justify your answer. [3 marks]
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