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.

 

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  • 3 months ago
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    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,...

    :)>

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

    S[4] = t[1] + t[2] + t[3] + t[4] = 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.

  • 3 months ago

    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?

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