Anonymous
Anonymous asked in Science & MathematicsMathematics · 2 years ago

# How to solve this series : 7 + 3 - 1 + (3/4) - (9/16) + (27/64) - ... ? The answer is 66/7 but I don't know how to solve it.?

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• [7 + 3 - 1] + [(3/4) - (9/16) + (27/64) - ... ]

= 9 + (3/7)

= 66/7

• You don't "solve" a series, but you can sum the series,

i.e., find out its total value.

I guess the series is meant to go on with alternating signs, and each term being (-3/4) times the previous term (even though this is not true of the first few terms.

Let's look at it as the sum of three distinct series:

(a) 7 + 3

(b) - 1 - (9/16) - (81/256) - (729/4096) - etc.

(c) (3/4) + (27/64) + (243/1024) + etc.

The sum of (a) is 10.

The sum of (c) is (3/4)/[1 - (9/16)] = (3/4)/(7/16) = 12/7.

The sum of (b) is -(4/3) times the sum of (c), so it's -16/7).

The sum of all three is 10 + 12/7 - 16/7 = 10 - 4/7 = 70/7 - 4/7 = 66/7.

• (3/4) - (9/16) + (27/64) - ...

S_∞ = (3/4)/[1+3/4] = 3/7

3/7 + 9 = 3/7 + 63/7

= 66/7

• (3/4) - (9/16) + (27/64) - ... = (3/4)/(1-(-3/4)) = 3/7

hence 7 + 3 - 1 + (3/4) - (9/16) + (27/64) - ... = ...