Given the geometric sequence that starts
7, 9, . . .
(a) Find the 7th term of the sequence. (Given that 7 is the first term.)
(b) Decide whether you can find the sum of all the terms of your sequence. If you can, find
that sum, otherwise, find the sum of the first 13 terms of the sequence.
r = 9/7
(a) a_9 = ar^8 = 7(9/7)^8
since r > 1 the infinite sum diverges.
S13 = (r^13 - 1)/(r-1) = ((9/7)^13 - 1)/(9/7 - 1) ≈ 88.32
the ratio (of consecutive terms) is ... 9/7
(a) the 7th term is ... 7 * (9/7)^6
(b) the ratio is not <1 , so the sequence does not converge
... the sum of the entire sequence is infinite
the sum of the 1st 13 terms ... (7 * {1 - [(9/7)^13]}) / [1 - (9/7)]
rats - forgot the a.
S13 = a(r^13 - 1)/(r-1) = 7*((9/7)^13 - 1)/(9/7 - 1) ≈ 618.24
To find the nth term of a geometric sequence, we need to determine the common ratio (r) first. In this case, we can find the common ratio by dividing any term by its previous term. Let's calculate the common ratio using the first and second terms:
Common ratio (r) = 9 / 7 = 1.285714...
(a) To find the 7th term of the sequence, we can use the formula for the nth term of a geometric sequence:
nth term = a * r^(n-1)
Given that the first term (a) is 7, and the common ratio (r) is 1.285714..., we can substitute these values into the formula:
7th term = 7 * 1.285714^(7-1)
= 7 * 1.285714^6
≈ 41.77946
Therefore, the 7th term of the sequence is approximately 41.77946.
(b) To determine if we can find the sum of all the terms, we need to check if the common ratio (r) is between -1 and 1 (excluding -1 and 1). In this case, the common ratio (1.285714...) is greater than 1, so we cannot find the sum of all terms.
However, we can find the sum of the first 13 terms of the sequence using the formula for the sum of a geometric sequence:
sum = a * (1 - r^n) / (1 - r)
Again, substituting the given values:
sum = 7 * (1 - 1.285714^13) / (1 - 1.285714)
≈ 198.45008
Therefore, the sum of the first 13 terms of the sequence is approximately 198.45008.