the 3th term of a GP is 63 while it 5th is 567.what is the sum of it's first seven terms
r^2 = 567/63 = 9
r = ±3
ar^2 = 63, so a = 7
and as usual,
S7 = a(r^7-1)/(r-1)
To find the sum of the first seven terms of the geometric progression (GP), we need to determine the common ratio (r) and the first term (a).
Given that the 3rd term (a3) is 63 and the 5th term (a5) is 567, we can set up two equations:
a * r^2 = 63 ----(1)
a * r^4 = 567 ----(2)
Dividing equation (2) by equation (1), we get:
(a * r^4) / (a * r^2) = 567 / 63
r^2 = 9
Taking the square root of both sides, we find that r = ±3.
Since a * 3^2 = 63 from equation (1), we can solve for a:
a * 9 = 63
a = 63 / 9
a = 7
Therefore, the first term (a) is 7 and the common ratio (r) is 3.
To find the sum of the first seven terms, we can use the formula for the sum of a geometric progression:
S = a * (1 - r^n) / (1 - r)
Plugging in the values, where n is the number of terms:
S = 7 * (1 - 3^7) / (1 - 3)
Simplifying,
S = 7 * (-3^7) / (-2)
S = (7 * 2187) / 2
S = 7659
Therefore, the sum of the first seven terms of the geometric progression is 7659.
To find the sum of the first seven terms of a geometric progression (GP), we need to determine the first term and the common ratio.
Let's denote the first term as 'a' and the common ratio as 'r'.
Given that the 3rd term is 63, we can write the following equation:
a * r^2 = 63 -- (Equation 1)
Similarly, the 5th term is 567:
a * r^4 = 567 -- (Equation 2)
To eliminate 'a', we can divide Equation 2 by Equation 1:
(a * r^4) / (a * r^2) = 567 / 63
Simplifying, we get:
r^2 = 9
Taking the square root of both sides, we find:
r = ±3
Now we have two possible values for 'r'. Let's examine each case separately:
Case 1: r = 3
Substituting r = 3 into Equation 1:
a * (3^2) = 63
Simplifying:
a * 9 = 63
a = 63 / 9
a = 7
Therefore, in this case, the first term is a = 7, and the common ratio is r = 3.
Case 2: r = -3
Substituting r = -3 into Equation 1:
a * (-3^2) = 63
Simplifying:
a * 9 = 63
a = 63 / 9
a = 7
Therefore, in this case, the first term is also a = 7, and the common ratio is r = -3.
Now that we have determined the first term (a = 7) and the common ratio (r = ±3), we can find the sum of the first seven terms using the formula for the sum of a geometric series:
Sum = a * (1 - r^n) / (1 - r)
Substituting the values we found, we get:
Sum = 7 * (1 - 3^7) / (1 - 3)
Calculating this expression will give us the sum of the first seven terms of the GP.