# A 100m cliff erodes by 2/7 of its height each year, after how many years is the height of the cliff 50 m. The initial height of the cliff is 100 m. 2/7 of its height is eroded every year. The height of the cliff remaining after every year is 5/7. The height of the cliff every year forms a geometric progression with common ratio 5/7 and initial value 100.

After n years,...

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The initial height of the cliff is 100 m. 2/7 of its height is eroded every year. The height of the cliff remaining after every year is 5/7. The height of the cliff every year forms a geometric progression with common ratio 5/7 and initial value 100.

After n years, the height of the cliff is given by 100*(5/7)^(n - 1)

100*(5/7)^(n - 1) = 50

=> (5/7)^(n - 1) = 1/2

take the log of both the sides

n - 1 = log 0.5/log(5/7)

=> n = 1 + 2.06

=> n = 3.06

The height of the cliff is 50 m after approximately 3 years.

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