27. Find the sum of 7 terms of the G.P 12, 6,
3……
A. 23.76
B. 27.36
C. 26.73
D. 23.81
To find the sum of the first 7 terms of a geometric progression (G.P), we can use the formula:
S_n = a(r^n - 1)/(r - 1)
Where:
- S_n is the sum of the first n terms
- a is the first term of the G.P
- r is the common ratio
- n is the number of terms
In this case, the first term (a) is 12 and the common ratio (r) is 1/2 (since each term is divided by 2 to get the next term). The number of terms (n) is 7.
Plugging these values into the formula, we get:
S_7 = 12(1/2^7 - 1)/(1/2 - 1)
= 12(1/128 - 1)/(-1/2)
= 12(1/128 - 1)/(-1/2)
= 12(-127/128)/(-1/2)
= 12 * (-127/128) * (-2/1)
= 24 * 127/128
= 3048/128
= 23.81
Therefore, the sum of the first 7 terms of the G.P 12, 6, 3... is approximately 23.81.
The correct option is D. 23.81.
To find the sum of the 7 terms of the given geometric progression (G.P), we need to use the formula for the sum of a geometric progression.
The formula to find the sum of a geometric progression is:
S = (a * (r^n - 1)) / (r - 1)
where:
S = sum of the terms
a = first term of the G.P
r = common ratio of the G.P
n = number of terms
Given that the first term (a) is 12, the common ratio (r) is 6/12 = 1/2, and the number of terms (n) is 7, we can substitute these values into the formula to find the sum.
S = (12 * ((1/2)^7 - 1)) / (1/2 - 1)
S = (12 * (1/128 - 1)) / (-1/2)
S = (12 * (1/128 - 128/128)) / (-1/2)
S = (12 * (-127/128)) / (-1/2)
S = (12 * (-127/128)) * (-2/1)
S = 2 * 12 * (127/128)
S = 24 * (127/128)
S = 3048/128
S = 23.8125
Therefore, the sum of the 7 terms of the given G.P is 23.8125, which is approximately equal to 23.81.
Therefore, the correct option is D. 23.81.