if the 2th and 4th term of a gp are 8 and 32 respectively. find the sum of the first 9th term
If the second and fourth terms of Geometric Progression (GP) is 8 and 32, it means they are two terms apart (4–2 = 2).
Since GPs follow the standard rule of multiplying the previous term by a fixed integer to get the subsequent term and that 8 and 32 are two terms apart, we simply have to find the quotient of 8 and 32 to the ratio of terms moved.
32/8 = 4 of which 4/2 terms = 2.
Therefore, the GP will be as follows 4, 8, 16, 32
The sum would then be 4 + 8 + 16 + 32 = 60
Using the familiar formulas you have been given:
"the 2th and 4th term of a gp are 8 and 32" ---> ar = 8 , ar^3 = 32
divide one equation by the other:
ar^3 / (ar) = 32/8
r^2 = 4
r = ± 2
if r = 2, a(2) = 8 ----> a = 4
if r = -2, a = -4
sum(9) = a(r^9 - 1)/(r-1)
for a=4,r=2 , Sum(9) = 4(2^9 -1)/1 = 2044
for a=-4, r = -2 , Sum(9) = -4((-2)^9 - 1)/(-2-1) = -684
Why did the geometric progression go to therapy? Because it was feeling a little "unbalanced"!
To find the sum of the first 9 terms in a geometric progression, we need to find the common ratio (r). We know that the 2nd term is 8 and the 4th term is 32.
So, let's find the common ratio first:
32/8 = 4
Now, we can use the sum of a geometric progression formula:
S_n = a(1 - r^n) / (1 - r)
Where:
S_n is the sum of the first n terms,
a is the first term,
r is the common ratio,
n is the number of terms.
In this case, a = 8 (2nd term), r = 4, and n = 9.
Plugging in the values:
S_9 = 8 * (1 - 4^9) / (1 - 4)
Calculating that out:
S_9 = 8 * (1 - 262144) / (1 - 4)
S_9 = 8 * (-262143) / (-3)
S_9 = 8 * 87381 = 699048
So, the sum of the first 9 terms is 699048! Keep in mind that this answer assumes the first term is not provided. If you have the first term, you can modify the calculation accordingly.
To find the sum of the first 9 terms of a geometric progression (GP), we need to first find the common ratio (r) and the first term (a).
Given that the 2nd term (a₂) is 8 and the 4th term (a₄) is 32, we can use these values to find the common ratio.
We know that the ratio between consecutive terms in a GP is constant, so we can find the common ratio (r) by dividing any term by its preceding term.
To find r:
r = a₄ / a₂
r = 32 / 8
r = 4
Now that we have the common ratio, we can find the first term (a) by dividing any term by the common ratio raised to its term number minus one.
To find a:
a = a₂ / r^(2-1)
a = 8 / 4
a = 2
Now that we know the first term (a = 2) and the common ratio (r = 4), we can find the sum of the first 9 terms of the GP using the formula for the sum of an n-term GP:
Sum of first n terms = a * (r^n - 1) / (r - 1)
Sum of first 9 terms = 2 * (4^9 - 1) / (4 - 1)
Calculating this expression, we get:
Sum of first 9 terms = 2 * (262144 - 1) / 3
Sum of first 9 terms = 2 * 262143 / 3
Sum of first 9 terms = 524286 / 3
Sum of first 9 terms ≈ 174762
Therefore, the sum of the first 9 terms of the geometric progression is approximately 174762.