For #1-4 write the \(nth\) term of the sequence in an explicit formula.
1) \(\Large\frac{5}{2},\Large\frac{7}{6},\Large\frac{3}{4},\Large\frac{11}{20},\Large\frac{13}{30},...\)
2) \(\Large\frac{1\cdot 2}{2},\Large\frac{2\cdot 3}{2},\Large\frac{3\cdot 4}{2},\Large\frac{4\cdot 5}{2},...\)
3) \(1,\ -3,\ 5,\ -7,\ 9...\)
4) \(1,\ 2,\ \frac{1}{3},\ 4,\ \frac{1}{5},\ 6,\ \frac{1}{7},...\)
5) Express the sum using summation notation in an explicit formula: \(3+-\Large\frac{3}{4}+\Large\frac{3}{16}+...\Large\frac{3}{4096}\).
6) Write out the sum, but do not evaluate: \(\Large\sum_\limits{k=1}^{5}\frac{(kx)^{2}+k}{2x}\).
7) Write the explicit formula by finding the values of \(a\), \(b\), and \(c\) when \({a}_{n}=a{n}^{2}+bn+c\) and the first three terms of the sequence are \(0, 11, 26, ...\)
Solution Bank
1) \(\Large\frac{5}{2},\Large\frac{7}{6},\Large\frac{3}{4},\Large\frac{11}{20},\Large\frac{13}{30},...\)
2) \(\Large\frac{1\cdot 2}{2},\Large\frac{2\cdot 3}{2},\Large\frac{3\cdot 4}{2},\Large\frac{4\cdot 5}{2},...\)
3) \(1,\ -3,\ 5,\ -7,\ 9...\)
4) \(1,\ 2,\ \frac{1}{3},\ 4,\ \frac{1}{5},\ 6,\ \frac{1}{7},...\)
5) Express the sum using summation notation in an explicit formula: \(3+-\Large\frac{3}{4}+\Large\frac{3}{16}+...\Large\frac{3}{4096}\).
6) Write out the sum, but do not evaluate: \(\Large\sum_\limits{k=1}^{5}\frac{(kx)^{2}+k}{2x}\).
7) Write the explicit formula by finding the values of \(a\), \(b\), and \(c\) when \({a}_{n}=a{n}^{2}+bn+c\) and the first three terms of the sequence are \(0, 11, 26, ...\)
Solution Bank