Equations of conic sections, Systems of Non-linear Equations illustrate a series, differentiate a series from a sequence Determine the first five terms of each defined sequence and give it's associated series 3^{n + 1} We have to find the first five terms of the given sequence and its associated series

Question
Conic sections
asked 2021-01-19
Equations of conic sections, Systems of Non-linear Equations illustrate a series, differentiate a series from a sequence Determine the first five terms of each defined sequence and give it's associated series \(3^{n\ +\ 1}\) We have to find the first five terms of the given sequence and its associated series

Answers (1)

2021-01-20
Given that, \(n^{th}\ \text{term of the sequence is}\ 3^{n\ +\ 1}.\) Take \(n = 0\) \(a_{n} = 3^{n\ +\ 1}\)
\(a_{0} = 3^{0\ +\ 1} = 3\) Therefore, the first term is 3. Take \(n = 1,\)
\(a_{1} = 3^{1\ +\ 1}\)
\(a_{1} = 3^{1\ +\ 1} = 3^{2} = 9\) The second term is 9 Take \(n = 2,\)
\(a2 = 3^{2\ +\ 1}\)
\(= 3^{3}\)
\(a_{2} = 27\) The third term is 27 For, \(n = 3,\)
\(a3 = 3^{3\ +\ 1}\)
\(= 3^{4}\)
\(a^{3} = 81\) Fourth term is 81 For, \(n = 4,\)
\(a4 = 3^{4\ +\ 1}\)
\(= 3^{5}\)
\(a^{4} = 243\) Fifth term is 243. Therefore, the first 5 terms of the sequence are \(3,\ 9,\ 27,\ 81,\ 243.\) Step 3 The given sequence is \(3,\ 9,\ 27,\ 81,\ 243.\) The series associated with the given sequence is \(\sum_{n=0}^{5}\ 3^{n\ +\ 1}=3\ +\ 9\ +\ 27\ +\ 81\ +\ 23\)
\(\sum_{n=0}^{5}\ 3^{n\ +\ 1}=363\)
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