The first term of geometric series is 2, the nth term is 486, and the sum of the n term is 728 ,find r and n.
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The first term of geometric series is 2, the nth term is 486, and the sum of the n term is 728 ,find
The first term of geometric series is 2, the nth term is 486, and the sum of the n term is 728 ,find r and n.
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Answer:
Hello! You could actually get both r and n with just the value of the first and nth term.
a₁ = 2
an = 486, which can be rewritten as:
486 = a_{1} r^{(n - 1)}486=a
1
r
(n−1)
Substitute the value of a₁:
486 = 2 r^{(n - 1)}486=2r
(n−1)
243 = r^{(n - 1)}243=r
(n−1)
3^{5} = r^{(n - 1)}3
5
=r
(n−1)
We can assume that r = 3
n - 1 = 5
n = 6
Substitute the value of a₁:
486 = 2 r^{(n - 1)}486=2r
(n−1)
243 = r^{(n - 1)}243=r
(n−1)
3^{5} = r^{(n - 1)}3
5
=r
(n−1)
−1)
To check if the values are correct, we may use the given value of Sn (which is 728)
Sn = a_{1} (\frac{1 - r^{n}}{1 - r})Sn=a
1
(
1−r
1−r
n
)
Sn = 2 (\frac{1 - 3^{6}}{1 - 3})Sn=2(
1−3
1−3
6
)
Sn = 2 (\frac{1 - 3^{6}}{-2})Sn=2(
−2
1−3
6
)
Sn = \frac{1 - 3^{6}}{-1}Sn=
−1
1−3
6
Sn = \frac{-728}{-1}Sn=
−1
−728
Sn = 728 (correct)
Summary:
r = 3
n = 6