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#geometric-series
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The sum of 3th to 7th term in a geometric series is 3267. Find the first term if the common ratio is 3?
Awa Kals
Mar 26
The sum of 3th to 7th term in a geometric series is 3267. Find the first term if the common ratio is 3?
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Mar 26
Let the first term of the geometric series be a and the common ratio be r.
The formula for the sum of the first n terms of a geometric series is:
S_n = a(r^n - 1)/(r - 1)
Given that the sum of the 3rd to 7th term is 3267, we can write:
S_7 - S_2 = 3267
a(r^7 - 1)/(r - 1) - a(r^2 - 1)/(r - 1) = 3267
a(r^7 - r^2) = 3267(r - 1)
Since the common ratio is 3, we can substitute r = 3 into the equation:
a(3^7 - 3^2) = 3267(3 - 1)
a(2187 - 9) = 6534
a(2178) = 6534
a = 6534/2178
a = 3
Therefore, the first term of the geometric series is 3.
The formula for the sum of the first n terms of a geometric series is:
S_n = a(r^n - 1)/(r - 1)
Given that the sum of the 3rd to 7th term is 3267, we can write:
S_7 - S_2 = 3267
a(r^7 - 1)/(r - 1) - a(r^2 - 1)/(r - 1) = 3267
a(r^7 - r^2) = 3267(r - 1)
Since the common ratio is 3, we can substitute r = 3 into the equation:
a(3^7 - 3^2) = 3267(3 - 1)
a(2187 - 9) = 6534
a(2178) = 6534
a = 6534/2178
a = 3
Therefore, the first term of the geometric series is 3.
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