For each of the following sequences: @ identify if the sequence is arithmetic, geometric or quadratic’. Justify your response. @ assuming the first item of each sequence is a1, give an expression for aj. (In other words, find a formula for the i-th term in the sequence). @ if the sequence is arithmetic or geometric, compute the sum of the first 10 terms in the sequence i 2,-12, 72, -432, 2592,... ii 9, 18, 31, 48, 69, 94,... iii 14, 11.5, 9, 6.5, 4, 1.5,...

Question
Polynomial arithmetic
asked 2020-12-17
For each of the following sequences: @ identify if the sequence is arithmetic, geometric or quadratic’. Justify your response. @ assuming the first item of each sequence is a1, give an expression for aj. (In other words, find a formula for the i-th term in the sequence). @ if the sequence is arithmetic or geometric, compute the sum of the first 10 terms in the sequence \(i 2,-12, 72, -432, 2592,...\)
\(ii 9, 18, 31, 48, 69, 94,...\)
\(iii 14, 11.5, 9, 6.5, 4, 1.5,...\)

Answers (1)

2020-12-18
Step 1 The series is \(2,−12,72,−432,2592,...\) check the ratio \(\frac{-12}{2} = -6\)
\(\frac{72}{-12} = -6\)
\(\frac{-432}{72} = -6\) this is geometrix series with common ratio r, where \(r = -6\) the sequence is geometric (b) the general term \(a_i = a_1 r^(i-1)\) where \(a_1\) is the first term , r is the common ratio. substitute the values \(a_i = 2(−6)^(i−1)\) hence, the expression for \(a_i is a_i = 2(−6)^{i -1}\) Step 2 the sum of first 10 terms is given by the formula \(\frac{a(1-r^n)}{1- r}\) substitute the values to get the sum of first 10 terms \(S_n = \frac{a(1-r^n)}{1- r}\)
\(= \frac{2(1-(-6)^n)}{1-(-6)}\)
\(= {2(1-(-6)^n)}{1+6}\)
\(= 17276050\) hence, the sum of first 10 terms is given by 17276050
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