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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.…
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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.

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User Saurabhshcs
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\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}

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User TrashyMcTrash
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