Awa Kals
Mar 26
How many terms in GP 4,3.6,3.24.... are needed so that the sum exceeds 35?
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ChatGPT
Mar 26
To find the number of terms needed in the geometric progression (GP) so that the sum exceeds 35, we can use the formula for the sum of the first n terms of a geometric progression:
S_n = a * (1 - r^n) / (1 - r)
Where:
S_n = sum of the first n terms
a = first term of the GP
r = common ratio of the GP
n = number of terms
In this case, the first term (a) is 4 and the common ratio (r) is 3.6 / 4 = 0.9.
We want to find the number of terms (n) needed so that the sum exceeds 35:
4 * (1 - 0.9^n) / (1 - 0.9) > 35
4 * (1 - 0.9^n) / 0.1 > 35
40 * (1 - 0.9^n) > 35
1 - 0.9^n > 35 / 40
1 - 0.9^n > 0.875
0.9^n < 0.125
n * log(0.9) < log(0.125)
n < log(0.125) / log(0.9)
n < 4.292
Therefore, we need at least 5 terms in the geometric progression so that the sum exceeds 35.
S_n = a * (1 - r^n) / (1 - r)
Where:
S_n = sum of the first n terms
a = first term of the GP
r = common ratio of the GP
n = number of terms
In this case, the first term (a) is 4 and the common ratio (r) is 3.6 / 4 = 0.9.
We want to find the number of terms (n) needed so that the sum exceeds 35:
4 * (1 - 0.9^n) / (1 - 0.9) > 35
4 * (1 - 0.9^n) / 0.1 > 35
40 * (1 - 0.9^n) > 35
1 - 0.9^n > 35 / 40
1 - 0.9^n > 0.875
0.9^n < 0.125
n * log(0.9) < log(0.125)
n < log(0.125) / log(0.9)
n < 4.292
Therefore, we need at least 5 terms in the geometric progression so that the sum exceeds 35.
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