1) Find the first \(5\) terms of the recursively defined sequence when \(a_1= -2: {a}_{n+1}=4{a}_{n}+3\).
2) Find the next \(3\) terms of the recursively defined sequence when \(a_1=-2, {a}_{2}=3\), and \({a}_{n+2}=n{a}_{n}-2{a}_{n+1}.\)
3) Find the first \(3\) terms of the recursively defined sequence when \(a_1=5, {a}_{2}=-3\) and \({a}_{n+2}=-2{a}_{n}{a}_{n+1}\).
4) If \(a_1=-4\) and \({a}_{n+1}=2{a}_{n}-5\), find \({a}_{4}\).
5) The second and fourth terms of a geometric sequence are \(2\) and \(6\). What is \(a_{10}\)?
6) Let \(a_1 = 97\), and for \(n > 1\), let \(a_n =\Large{\frac{n}{x_n−1}}\). Calculate the product of \(a_1a_2\cdot\cdot\cdot a_8\).
7) A sequence \({a_n}\) satisifies \(a_2=2a_1,\;a_{n+2} - 4_{n+1} + 3a_n = 0\) where \(n\) is a positive integer. If \(a_5=287\), what is the value of \(a_4?\) (from brilliant.org)
8) If \({b_n}\) is a geometric progression where all terms are positive and \(b_3=\frac{1}{4}\) and \(b_7=\frac{1}{64}\), what is the smallest integer \(n\) such that \(b_n < \frac{1}{1000}\)?
9) A sequence \({b_n}\) with \(b_n>0\) for all \(n>1\) satisfies \(\Large\frac{b_{n+1}-b_n}{b_{n+1}+b_n}=\frac{1}{2}\) If \(b_1=2\), what is the value of \(b_5\)?
10) If \(f(x)=3x-11\), find: \(f(f(f(2x-1)))\).
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2) Find the next \(3\) terms of the recursively defined sequence when \(a_1=-2, {a}_{2}=3\), and \({a}_{n+2}=n{a}_{n}-2{a}_{n+1}.\)
3) Find the first \(3\) terms of the recursively defined sequence when \(a_1=5, {a}_{2}=-3\) and \({a}_{n+2}=-2{a}_{n}{a}_{n+1}\).
4) If \(a_1=-4\) and \({a}_{n+1}=2{a}_{n}-5\), find \({a}_{4}\).
5) The second and fourth terms of a geometric sequence are \(2\) and \(6\). What is \(a_{10}\)?
6) Let \(a_1 = 97\), and for \(n > 1\), let \(a_n =\Large{\frac{n}{x_n−1}}\). Calculate the product of \(a_1a_2\cdot\cdot\cdot a_8\).
7) A sequence \({a_n}\) satisifies \(a_2=2a_1,\;a_{n+2} - 4_{n+1} + 3a_n = 0\) where \(n\) is a positive integer. If \(a_5=287\), what is the value of \(a_4?\) (from brilliant.org)
8) If \({b_n}\) is a geometric progression where all terms are positive and \(b_3=\frac{1}{4}\) and \(b_7=\frac{1}{64}\), what is the smallest integer \(n\) such that \(b_n < \frac{1}{1000}\)?
9) A sequence \({b_n}\) with \(b_n>0\) for all \(n>1\) satisfies \(\Large\frac{b_{n+1}-b_n}{b_{n+1}+b_n}=\frac{1}{2}\) If \(b_1=2\), what is the value of \(b_5\)?
10) If \(f(x)=3x-11\), find: \(f(f(f(2x-1)))\).
Solution Bank