# MATH EXPONENTIAL QUESTION?

For a geometric series S₂/S₄ = 1/17, determine the first three terms of the series if the first term is 3. series , determine the first three terms of the series if the first term is 3.

Relevance

Sₙ = a(1 - rⁿ)/(1 - r)

so, S₂ = 3(1 - r²)/(1 - r) and S₄ = 3(1 - r⁴)/(1 - r)

Hence, S₂/S₄ = (1 - r²)/(1 - r⁴)

i.e. (1 - r²)/(1 - r²)(1 + r²)

or, 1/(1 + r²) = 1/17

so, 1 + r² = 17

=> r² = 16

Then, r = 4 or -4

Using r = 4 we get:

3, 12, 48, 192,...

Using r = -4 we get:

3, -12, 48, -192,...

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• S = t + t = ar + ar^2 = ar * (1 + r)

S = t + t + t + t = ar + ar^2 + ar^3 + ar^4 = ar * (1 + r + r^2 + r^3)

ar = 3

3 * (1 + r) / (3 * (1 + r + r^2 + r^3)) = 1/17

(1 + r) / (1 + r + r^2 + r^3) = 1/17

17 * (1 + r) = 1 + r + r^2 + r^3

17 * (1 + r) = 1 + r + r^2 * (1 + r)

17 * (1 + r) = (1 + r) * (1 + r^2)

1 + r = 0

r = -1

That's a possible value for r, but let's not get too attached to it yet

17 = 1 + r^2

16 = r^2

r = -4 , 4

3 * (1 + (-1)) = 3 * 0 = 0

3 * (1 - 1 + 1 - 1) = 3 * 0 = 0

0/0 is undefined.  r = -1 doesn't work.

3 * (1 - 4) = 3 * (-3) = -9

3 * (1 - 4 + 16 - 64) = 3 * (17 - 68) = 3 * (-51) = -153

-9/-153 = 1/17

r = -4 works

3 * (1 + 4) = 3 * 5 = 15

3 * (1 + 4 + 16 + 64) = 3 * (5 + 80) = 3 * 85 = 255

15/255 = 1/17

r = 4 works as well.

3 , -12 , 36

3 , 12 , 36

Those are the first 3 terms of each possible sequence.

• Is S₂ the second term? or the sum of the first two terms?

Is S₄ the fourth term? or the sum of the first four terms?