find the following gp with sequence 2,8,32,128
find the sum of its first 8 term
find the sum of its first 13 terms
as always,
Sn = a(r^n - 1)/(r-1)
To find the common ratio (r) of the given geometric progression (GP), we divide any term by its previous term.
Given terms:
a₁ = 2
a₂ = 8
a₃ = 32
a₄ = 128
Let's find the value of r:
r = a₂ / a₁
= 8 / 2
= 4
Hence, the common ratio (r) of the GP is 4.
To find the sum of the first 'n' terms of a GP, we use the formula:
Sn = a₁ * (r^n - 1) / (r - 1)
Now let's find the sum of the first 8 terms:
n = 8
a₁ = 2
r = 4
Sn = 2 * (4^8 - 1) / (4 - 1)
= 2 * (65536 - 1) / 3
= 2 * 65535 / 3
= 131070 / 3
≈ 43690
Hence, the sum of the first 8 terms of the GP is approximately 43690.
Next, let's find the sum of the first 13 terms:
n = 13
a₁ = 2
r = 4
Sn = 2 * (4^13 - 1) / (4 - 1)
= 2 * (67108864 - 1) / 3
= 2 * 67108863 / 3
= 134217726 / 3
≈ 44739242
Hence, the sum of the first 13 terms of the GP is approximately 44739242.
To find the common ratio (r) in a Geometric Progression (GP), we divide any term by its previous term.
First, let's find the common ratio (r):
Dividing the second term (8) by the first term (2):
r = 8/2 = 4
Dividing the third term (32) by the second term (8):
r = 32/8 = 4
Dividing the fourth term (128) by the third term (32):
r = 128/32 = 4
Hence, the common ratio (r) is 4.
Now that we have the common ratio, we can find the terms of the GP using the formula:
aₙ = a₁ * r^(n-1)
where a₁ is the first term, r is the common ratio, and n is the term number.
Using this formula, we can find the terms of the GP:
The first term (a₁) is 2.
The second term (a₂) = 2 * 4^(2-1) = 2 * 4 = 8
The third term (a₃) = 2 * 4^(3-1) = 2 * 16 = 32
The fourth term (a₄) = 2 * 4^(4-1) = 2 * 64 = 128
Now, let's find the sum of the first 8 terms of the GP:
The sum of the first 8 terms of a GP can be calculated using the formula:
Sₙ = a₁ * (r^n - 1) / (r - 1)
where Sₙ is the sum of the first n terms.
Using this formula, we can find the sum of the first 8 terms:
S₈ = 2 * (4^8 - 1) / (4 - 1)
= 2 * (65536 - 1) / 3
= 2 * 65535 / 3
= 131070 / 3
≈ 43690
So, the sum of the first 8 terms is approximately 43690.
Now, let's find the sum of the first 13 terms of the GP:
Using the same formula, we can find the sum of the first 13 terms:
S₁₃ = 2 * (4^13 - 1) / (4 - 1)
= 2 * (67108864 - 1) / 3
= 2 * 67108863 / 3
= 134217726 / 3
≈ 44739242
So, the sum of the first 13 terms is approximately 44739242.