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Question 1167088: The differences between the terms of the geometric sequence are known: '[a2-a1=-6,a3-a2=12' Find the sum of the first ten terms of that sequence.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! it looks like the differences are doubling every time and the signs are reversing every time.
A2 minus A1 = -6
A3 minus A2 = 12
since this is a geometric series, the formula is An = A1 * r^(n-1), where r is the common ratio.
this means that:
A2 = A1 * r
A3 = A1 * r^2
we also know that:
A2 - A1 = -6
A3 - A2 = 12
since A2 = A1 * r, then A2 - A1 becomes A1 * r - A1 = -6
since A3 = A1 * r^2, then A3 - A2 = 12 becomes A1 * r^2 - A1 * r = 12
factor out the A1 in both equations to get:
A1 * (r - 1) = -6
A1 * (r^2 - r) = 12
solve for A1 in both equations to get:
A1 = -6 / (r - 1)
A1 = 12 / (r^2 - r)
since they are both equal to A1, we get:
-6 / (r - 1) = 12 / (r^2 - r)
cross multiply to get:
-6 * (r^2 - r) = 12 * (r - 1)
divide both sides of this equation by (r - 1) to get:
-6 * (r^2 - r) / (r - 1) = 12
divide both sides of this equation by -6 to get:
(r^2 - r) / (r - 1) = 12 / -6 = -2
factor the numerator on the left side of the equation to get:
r * (r - 1) / (r - 1) = -2
simplify the equation to get:
r = -2
you now know that the common ratio = -2.
knowing that, we get:
A2 - A1 = -6 becomes A1 * r - A1 = -6 which becomes A1 * -2 - A1 = -6 which becomes:
-2 * A1 - A1 = -6
combine like terms to get -3 * A1 = -6
solve for A1 to get A1 = -6 / -3 = 2
since we now know the value of A1, we get:
A1 = 2
A2 = 2 * -2 = -4
A3 = 2 * (-2)^2 = 2 * 4 = 8
we have:
A1 = 2
A2 = -4
A3 = 8
A2 - A1 = -4 - 2 = -6
A3 - A2 = 8 - (-4) = 12
it looks like we have A1 and the common ratio of -2 nailed down pretty good and we can proceed with finding the first ten terms of our sequence.
we get:
A1 = 2
A2 = 2 * -2 = -4
A3 = -4 * -2 = 8
A4 = 8 * -2 = -16
A5 = -16 * -2 = 32
A6 = 32 * -2 = -64
A7 = -64 * -2 = 128
A8 = 128 * -2 = -256
A9 = -256 * -2 = 512
A10 = 512 * -2 = -1024
we add all ten terms to get:
the sum of the first 10 terms of the sequence = -684 !!!!!
we can also use the sum of the terms of a geometric sequence to see if we get the same value.
that formula is:
Sn = A1 * (r - r^n) / (1 - r)
when n = 10 and A1 = 2, this formula becomes:
S10 = 2 * (-2 - (-2)^10) / (1 - (-2))
simplify this formula to get:
S10 = 2 * (-2 - 1024) / (-3)
simplify further to get:
S10 = -2 * -1026 / -3
solve for S10 to get:
S10 = -684 !!!!!
the formula gives the same answer as manually summing up the terms.
this confirms that the sum formula works and that the sum of the first ten terms of the sequence is -684.
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