# The first term of an infinite geometric series is 48. The ratio of the series is 0.6. (a) Find the third term of the series. (b) Find the sum to infinity. (c) The nth term of the series is u_n. Find the value of the sum from n=4 to infinity of u_n.

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Note here: u_n indicates u subscript n.

(a) u_1 = 48 and the ratio, r = 0.6

Using a calculator, u_2 = 48 x 0.6 = 28.8

u_3 = 28.8 x 0.6 = 17.28

(b) We have the known result that the sum to infinity of a geometric series is a/(1-r) where a is the first term and r is the common ratio.

Therefore, the sum to infinity here is 48/(1-0.6) = 48/0.4 = 120

(c) We now want the sum from the fourth term to infinity. We can use the same formula as before, but replacing the first term which we called a with the fourth term of the sequence.

Calculating the fourth term: u_4 = 17.28 x 0.6 = 10.368

Therefore, our sum is equal to 10.368/(1-0.6) = 10.368/0.4 = 25.92

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