Question

The common ratio of a geometric progression is 0.99. Express the sum of the first 100 terms as a
percentage of the sum to infinity, giving your answer correct to 2 significant figures. [5 marks)

Answer 1

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The two values that we consider important when talking about geometric series, are the initial term (a) and the common ratio (r). We are given r = 0.99We know the formula for the sum of the first n terms of the progression to be: a*(1-rn)/(1-r) (the formula can be used without proof, although it's helpful to understand how we reach this)We don't know the value of a, so we will keep expressing it like this, but we replace the values that we know so that we can get the percentae that we are looking for: r < 1 => r^n converges to 0 as n increases therefore, our expression will look like: [a*(1-0.99100)/(1-0.99)]/[a*(1-0)/(1-0.99)] the a's cancel out, so we can simplify everything to: (1-0.366)/1 which further simplifies to: 0.634/1 = 0.634Because we are asked to give our result as a percentage, we will write 0.634 as being 63% (percentages are generally given to 2 s.f. unless specified otherwise)

Answer 2

Answer:

The two values that we consider important when talking about geometric series, are the initial term (a) and the common ratio (r). We are given r = 0.99We know the formula for the sum of the first n terms of the progression to be: a*(1-rn)/(1-r) (the formula can be used without proof, although it's helpful to understand how we reach this)We don't know the value of a, so we will keep expressing it like this, but we replace the values that we know so that we can get the percentae that we are looking for: r < 1 => r^n converges to 0 as n increases therefore, our expression will look like: [a*(1-0.99100)/(1-0.99)]/[a*(1-0)/(1-0.99)] the a's cancel out, so we can simplify everything to: (1-0.366)/1 which further simplifies to: 0.634/1 = 0.634Because we are asked to give our result as a percentage, we will write 0.634 as being 63% (percentages are generally given to 2 s.f. unless specified otherwise)

Step-by-step explanation:

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