Find the 8th term and the sum of the series 3,6,12,_____ up to 8 terms
a=3
r = 2
a8 = ar^7
S8 = a(r^8 - 1)/(r-1)
So plug and chug
What is the sum of the first 8 terms of a geometric sequence given it's a¹=2 and r=–4?
That is geometric progression where:
a1 = initial value
r = common ratio
Common ratio:
r = a2 / a1 = 6 / 3 = 2
r = a3 / a2 = 12 / 6 = 2
In this case a1 = 3 , r = 2
a1 = 3
a2 = a1 ∙ r = 3 ∙ 2 = 6
a3 = a2 ∙ r = 6 ∙ 2 = 12
a4 = a3 ∙ r = 12 ∙ 2 = 24
a5 = a4 ∙ r = 24 ∙ 2 = 48
a6 = a5 ∙ r = 48 ∙ 2 = 96
a7 = a6 ∙ r = 96 ∙ 2 = 192
a8 = a7 ∙ r = 192 ∙ 2 = 384
nth partial sum of a geometric sequence:
Sn = a1 ( 1 - r ⁿ ) / ( 1 - r )
In this case:
S8 = a1 ( 1 - r ⁸ ) / ( 1 - r )
S8 = 3 ∙ ( 1 - 2 ⁸ ) / ( 1 - 2 )
S8 = 3 ∙ ( 1 - 256 ) / ( - 1 )
S8 = 3 ∙ ( - 255 ) / ( - 1 )
S8 = 3 ∙ 255 = 765
You can chect that:
3 + 6 + 12 + 24 + 48 + 96 + 192 + 384 = 765
To find the 8th term and the sum of the series, we need to determine the pattern of the given sequence. Looking closely at the given terms: 3, 6, 12, we can observe that each term is obtained by multiplying the previous term by 2.
Starting with the first term, which is 3:
3 * 2 = 6 (second term)
6 * 2 = 12 (third term)
12 * 2 = 24 (fourth term)
24 * 2 = 48 (fifth term)
48 * 2 = 96 (sixth term)
96 * 2 = 192 (seventh term)
192 * 2 = 384 (eighth term)
Therefore, the 8th term is 384.
To find the sum of the series up to the 8th term, we can use the formula for the sum of a geometric series:
S = a * (r^n - 1) / (r - 1)
Where:
S is the sum of the series
a is the first term
r is the common ratio
n is the number of terms
In our case, a = 3, r = 2, and n = 8.
Using the formula:
S = 3 * (2^8 - 1) / (2 - 1)
Simplifying further:
S = 3 * (256 - 1) / 1
S = 3 * 255
S = 765
Therefore, the sum of the series up to 8 terms is 765.