.In a geometric sequence,T6=-243 and T3=72
Determine the sum of the first 10
terms of the sequence.
To find the sum of the first 10 terms of a geometric sequence, we first need to find the common ratio (r) and the first term (T1).
We are given that T6 = -243 and T3 = 72. We can use these values to find the common ratio:
T6 = T3 * r^3
-243 = 72 * r^3
r^3 = -243 / 72
r^3 = -3.375
r = -3^(3/3) = -3
Now we can find the first term (T1) using T3:
T3 = T1 * r^(3-1)
72 = T1 * (-3)^2
72 = T1 * 9
T1 = 8
Therefore, the first term (T1) is 8 and the common ratio (r) is -3.
Now we can find the sum of the first 10 terms of the sequence using the formula for the sum of a finite geometric series:
S = T1 * (r^n - 1) / (r - 1)
= 8 * (-3^10 - 1) / (-3 - 1)
= 8 * (59049 - 1) / (-4)
= 8 * 59048 / -4
= -118096
So, the sum of the first 10 terms of the geometric sequence is -118096.