the third term of a G.P is 63 its fifth term is 567, what is the sum of its first three term?
The third term of a g.p is 63 while its fifth terms is 567.what is the sum of its first seven term
To find the sum of the first three terms of a geometric progression (G.P.), we first need to determine the common ratio (r).
Given that the third term (a3) of the G.P. is 63, and the fifth term (a5) is 567, we can set up the following equations:
a3 = a1 * r^2 (Equation 1)
a5 = a1 * r^4 (Equation 2)
We can divide Equation 2 by Equation 1, which gives:
a5 / a3 = a1 * r^4 / (a1 * r^2)
567 / 63 = r^2
Simplify the equation:
9 = r^2
Taking the square root of both sides:
r = ±3
Since geometric progressions only have positive common ratios, we take r = 3.
Now that we have the common ratio, and since we want to find the sum of the first three terms, we have:
a1 = a3 / r^2
= 63 / (3^2)
= 63 / 9
= 7
So, the first term (a1) is 7, and the common ratio (r) is 3. Now we can compute the sum of the first three terms (S3):
S3 = a1 + a2 + a3
Substituting the values:
S3 = 7 + 7r + 7r^2
Using the common ratio (r = 3):
S3 = 7 + 7(3) + 7(3^2)
= 7 + 21 + 63
= 91
Therefore, the sum of the first three terms of the geometric progression is 91.