Given the sequence:1,-1/3,1/9,-1/27,... Find a) the fifteenth term. b) the sum of the 5th term to be 10th term
Mathematics · Middle School · Tue Nov 03 2020
Answered on
Given the sequence:
1, - ⅓, 1/9, - 1 / 27
Determine the fifteenth term.
Determine the sum of the 5th term to the 10th term
Solution:
In order to give the answers for the conditions that were stated, we can manually solve each term. To do so, we can see that the fraction is being multiplied by -⅓ since the value is decreasing at a rate of ⅓ and it varies from negative to positive, this may only mean that ⅓ is negative. Now to find up to the fifteenth term, since we are also tasked to find the sum of the 5th and 10th terms, then we can manually list the values starting from the 5th term up until the 10th term. For the 15th term, we will use the formula for geometric sequence denoted as,
Formula for Geometric sequence:
An = a1.( r )^(n-1)
A15 = 1.(-⅓)^(15-1)
A15 = 1.(-⅓)^(14-1)
A15 = 1 / 478969
List the 5th term up until the 10th term
5th term = - 1 / 27 (-⅓)
5th term = 1/81
6th term = 1/81 (-⅓)
6th term = - 1 / 243
7th term = -1 / 243 (-⅓)
7th term = 1/729
8th term = 1/729 (-⅓)
8th term = - 1 / 2187
9th term = - 1/2187 (-⅓)
9th term = 1 / 6561
10th term= 1/6561(-⅓)
10th term = -1/19683
Add all the terms together. To add, we can convert each one to decimal in order to compute easily.
= (1/81) + (- 1 /243) + (1/729) + (-1 / 2187) + (1/6561) + (-1/19683)
=0.00924655 or 182/19683
Final answer:
15th term: A15 = 1 / 478969
Sum of the 5th term to the 10th term:
=0.00924655 or 182/19683