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Question 1007600: Please help I have an idea but dont understad. Thanks
1. Find first 5 susesions of:
an=2(1/3) (fraccion) exponent n-1
2.
250
∑ 4 = (4) + (-4) + 1 + 4 + 4 =
n=1
3.
∑5 (4/5) Exponent n
n=1
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! an = 2 * (1/3)^(n-1)
this is based on the general geometric series formula of:
an = a1 * r^(n-1)
you can calculate the first 5 terms directly and then add them up.
you will get:
a1 = 2 * (1/3)^0 = 2 * 1 = 2
a2 = 2 * (1/3)^1 = 2 * 1/3 = 2/3
a3 = 2 * (1/3)^2 = 2 * 1/9 = 2/9
a4 = 2 * (1/3)^3 = 2 * 1/27 = 2/27
a5 = 2 * (1/3)^4 = 2 * 1/81 = 2/81
make them all have a denominator of 81 and you will get:
a1 = 162/81
a2 = 54/81
a3 = 18/81
a4 = 6/81
a5 = 2/81
add them up and you get:
s5 = 242/81.
alternatively, you could use the sum of a geometric series formula.
that formula is:
sn = a1 * (1 - r^n) / (1-r)
in your problem:
a1 = 2
r = 1/3
n = 5
the formula becomes:
s5 = 2 * (1 - (1/3)^5) / (1-1/3) which becomes:
s5 = 2 * (1 - 1/243) / (2/3) which becomes:
s5 = 2 * (242/243) * (3/2) which becomes:
s5 = 484/243 * 3/2 which becomes:
s5 = 1452/486 which is equivalent to:
s5 = 242/81 which is the same result as your got above.
obviously, if you know the formula, then your job becomes easy.
in this case you could have just added them up manually, but if the number of terms in the series was much greater, like about 50 or 100, and you have to do it manually rather than using a computer, then the formula becomes indispensable.
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