To find the common ratio, we can use the formula for the sum of an infinite geometric series:
S = a / (1 - r)
where S is the sum of the series, a is the first term, and r is the common ratio.
In this case, S = 889 and a = 7. So we have:
889 = 7 / (1 - r)
Simplifying, we get:
1 - r = 7 / 889
r = 1 - (7 / 889) = 1 - 0.0079 ≈ 0.9921
To find the number of terms, we can use the formula for the sum of a finite geometric series:
S = a * (1 - r^n) / (1 - r)
where S is the sum of the series, a is the first term, r is the common ratio, and n is the number of terms.
In this case, S = 889, a = 7, and r ≈ 0.9921. Substituting these values, we get:
889 = 7 * (1 - 0.9921^n) / (1 - 0.9921)
1 - 0.9921^n = (1 - 0.9921) / 7
0.9921^n = 1 - (1 - 0.9921) / 7
0.9921^n = 1 - 0.0079 / 7
0.9921^n ≈ 0.9989
Taking the logarithm of both sides, we get:
n * ln(0.9921) ≈ ln(0.9989)
n ≈ ln(0.9989) / ln(0.9921)
n ≈ 456.23
Since the number of terms must be a whole number, we round n down to 456.
Therefore, the common ratio is approximately 0.9921 and the number of terms is 456.
In a G.P the first term is 7, the last time is 448 and the sum to n terms is 889. The common ratio and the number of terms are
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