Register

A geometric sequence has all positive terms. The sum of the first two terms is 15 and the sum to infinity is 27.a Find the value of the commonratio.b Hence, find the first term.…

A geometric sequence has all positive terms. The sum of the first two terms is 15 and the sum to infinity is 27.a Find the value of the commonratio.b Hence, find the first term.…
60.7k views
2 votes
A geometric sequence has all positive terms. The sum of the first two terms is 15 and the sum to infinity is 27.a Find the value of the commonratio.b Hence, find the first term.

User Saurabhshcs
by
8.3k points

1 Answer

5 votes

\begin{gathered} a)S_2=\text{ the sum of the first two terms} \\ S_2=a_1+a_2=15 \\ S_2=a_1((r^2-1)/(r-1)) \\ r=\text{ratio} \\ 15=a_1((r^2-1)/(r-1))\text{ , but} \\ r^2-1=(r+1)(r-1),\text{ hence} \\ 15=a_1(((r+1)(r-1))/(r-1)) \\ 15=a_1(r+1) \\ \text{Sum to infinity} \\ S_(\infty)=(a_1)/(1-r) \\ 27=(a_1)/(1-r) \\ 27(1-r)=a_1\text{ (2)} \\ U\sin g\text{ (2) in (1)} \\ 15=27(1-r)(r+1) \\ (1-r)(r+1)=r+1-r^2-r \\ (1-r)(r+1)=1-r^2 \\ 15=27(1-r^2) \\ 15=27-27r^2 \\ 15+27r^2=27 \\ 27r^2=27-15 \\ 27r^2=12 \\ r^2=(12)/(27) \\ r^2=(4)/(9) \\ √(r^2)=\sqrt{(4)/(9)} \\ r=(2)/(3) \\ \text{The value of the common ratio is }(2)/(3) \\ b)For\text{ }a_1 \\ 27(1-r)=a_1 \\ a_1=27(1-(2)/(3)) \\ a_1=27((1)/(3)) \\ a_1=9 \\ The\text{ value of the first term is 9} \\ \end{gathered}

User TrashyMcTrash
by
7.8k points
← Prev Question Next Question →

Related questions

Ask a Question
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories

Other Questions

How do you can you solve this problem 37 + y = 87; y =
What is .725 as a fraction
How do you estimate of 4 5/8 X 1/3
Link Copied!