The third term of a gp is 63 while its 5th term is 567. What is the sum of its 1st seven terms.
a r^(n-1)
third a r^2 = 63
fifth a r^4 = 567
r^4/r^2 = r^2 = 567/63 = 9
so r = 3
a r^2 = 9 a = 63
so a = 7
so
Tn = 7 * 3^(n-1)
I think you can take it from there.
Why did the math book go to the doctor? Because it had too many problems!
To find the common ratio (r) of the geometric progression (GP), we can divide the 5th term by the 3rd term:
r = 567 / 63 = 9
Next, we can find the first term (a) using the formula:
a = 3rd term / (r^2)
a = 63 / (9^2) = 63 / 81 = 7/9
Now, we can find the sum of the first seven terms (S7) using the formula for the sum of a geometric series:
S7 = a * (r^7 - 1) / (r - 1)
S7 = (7/9) * (9^7 - 1) / (9 - 1)
Simplifying this expression gives us:
S7 = (7/9) * (387420489 - 1) / 8
S7 = (7/9) * (387420488) / 8
S7 ≈ 30240
To find the sum of the first seven terms of a geometric progression (GP), we need to first find the common ratio (r).
We are given the third term (a₃) as 63 and the fifth term (a₅) as 567.
We can use these values to find r.
The general formula for the nth term of a GP is:
an = a₁ * r^(n-1),
where a₁ is the first term.
Using this formula, we can set up two equations using the given information:
a₃ = a₁ * r^(3-1) = a₁ * r² = 63,
and
a₅ = a₁ * r^(5-1) = a₁ * r⁴ = 567.
Dividing these two equations, we have:
(a₁ * r⁴) / (a₁ * r²) = 567 / 63,
which simplifies to:
r² = 9.
Taking the square root of both sides, we get:
r = ±3.
Since we are dealing with a geometric progression, the common ratio cannot be negative. Therefore, r = 3.
Now that we have the common ratio, we can find the first term (a₁) by substituting the value of r into one of the original equations. Let's use the equation a₃ = a₁ * r² = 63:
a₁ * 3² = 63,
9a₁ = 63,
a₁ = 7.
So, the first term (a₁) is 7 and the common ratio (r) is 3.
To find the sum of the first seven terms, we will use the formula:
Sn = a₁ * (r^n - 1) / (r - 1),
where Sn is the sum of the first n terms.
Plugging in the known values:
n = 7,
a₁ = 7,
r = 3,
we have:
S₇ = 7 * (3^7 - 1) / (3 - 1).
Evaluating this expression gives us:
S₇ = 7 * (2187 - 1) / 2,
S₇ = 7 * 2186 / 2,
S₇ = 7631.
Therefore, the sum of the first seven terms is 7631.