The 5 term of a G.P is 63 while its 7 term is 567. What is the sum of its first eight terms?
Let's call the first term of the geometric progression a, and the common ratio between terms r.
From the given information, we know that:
a * r^4 = 63 ----- (1)
a * r^6 = 567 ----- (2)
Dividing equation (2) by equation (1), we get:
(r^6) / (r^4) = 567 / 63
r^2 = 9
Taking the square root of both sides, we find:
r = 3
Plugging the value of r into equation (1), we get:
a * 3^4 = 63
a * 81 = 63
a = 63 / 81
a = 7 / 9
Now that we know the first term (a = 7/9) and the common ratio (r = 3) for the geometric progression, we can calculate the sum of the first eight terms using the formula:
S = a * (1 - r^n) / (1 - r)
where a is the first term, r is the common ratio, and n is the number of terms.
Plugging in the values, we have:
S = (7/9) * (1 - 3^8) / (1 - 3)
Simplifying:
S = (7/9) * (-6561) / (-2)
S = 2293 / 3
Therefore, the sum of the first eight terms of the geometric progression is 2293 / 3.