A question on Geometric Sequence
Problem
The sum of the 1st and 2nd terms of a geometric progression is 50 and the sum of the 2nd and 3rd terms is 30. Find the sum to infinity.
Solution
Sometimes, it is better using the term formula and manipulate algebra to obtain a simpler look for the equation. From the given information, we have equation (1) \( a + ar = 50 \implies a(1+r) = 50 \) and equation (2) \( ar + ar^2 = 30 \implies ar(1+r) = 30 \). However, we do know that \(a(1+r) = 50\) and judging from second equation, we have \( r\times 50 = 30 \implies \displaystyle r = \frac{3}{5} \). From here, you can find \( a = \displaystyle \frac{125}{4} \). Once you have a, you can use the formula for sum to infinity \( \displaystyle S_{\infty} = \frac{a}{1-r} =\frac{625}{8}\)
The sum of the 1st and 2nd terms of a geometric progression is 50 and the sum of the 2nd and 3rd terms is 30. Find the sum to infinity.
Solution
Sometimes, it is better using the term formula and manipulate algebra to obtain a simpler look for the equation. From the given information, we have equation (1) \( a + ar = 50 \implies a(1+r) = 50 \) and equation (2) \( ar + ar^2 = 30 \implies ar(1+r) = 30 \). However, we do know that \(a(1+r) = 50\) and judging from second equation, we have \( r\times 50 = 30 \implies \displaystyle r = \frac{3}{5} \). From here, you can find \( a = \displaystyle \frac{125}{4} \). Once you have a, you can use the formula for sum to infinity \( \displaystyle S_{\infty} = \frac{a}{1-r} =\frac{625}{8}\)
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