Sample quiz on recursion formulas
Main home here.

1. A recursion formula is given by $t_{n}=t_{n-1}+3,\quad t_{1}=0$. Find $t_{4}$.
   
   $13$
   $12$
   $9$
   $6$
2. A recursion formula is given by $t_{n}=-2\times t_{n-1},\quad t_{1}=1$. Find $t_{5}$.
   
   $16$
   $8$
   $-8$
   $-16$
3. Find a recursion formula for the sequence $1,3,5,7,9,\cdots$.
   
   $t_{n}=t_{n+1}+2,\quad t_{1}=1$
   $t_{n}=t_{n-1}+2,\quad t_{1}=1$
   $t_{n}=t_{n-1}-2,\quad t_{1}=1$
   $t_{n}=t_{n-2}-2,\quad t_{1}=1$
   
4. Find a recursion formula for the sequence $-1,3,-9,27,-81,\cdots$.
   
   $t_{n}=-3\times t_{n+1},\quad t_{1}=-1$
   $t_{n}=-3\times t_{n-1},\quad t_{1}=-1$
   $t_{n}=-3\times t_{n-2},\quad t_{1}=-1$
   $t_{n}=-3\times t_{n+2},\quad t_{1}=-1$
5. A recursive formula is given by $t_{n}=a\times t_{n-1}+b,\quad t_{1}=k$. For what values of $a,b,k$ can the sequence be geometric?
   
   $a=2,~b=0,~k=0$
   $a=1,~b=0,~k=1$
   $a=2,~b=0,~k=1$
   $a=1,~b=0,~k=0$
6. For what values of $a,b,k$ can the recursive formula $t_{n}=a\times t_{n-1}+b,\quad t_{1}=k$ yield an arithmetic sequence?
   
   $a=2,~b=0,~k=1$
   $a=2,~b=0,~k=2$
   $a=1,~b=2,~k=1$
   $a=2,~b=0,~k=3$
7. Find a recursive formula for the sequence $-2,-5,-8,-11,-14,\cdots$.
   
   $t_{n}=t_{n+1}-3,\quad t_{1}=-2$
   $t_{n}=t_{n-1}+3,\quad t_{1}=-2$
   $t_{n}=t_{n-2}-3,\quad t_{1}=-2$
   $t_{n}=t_{n-1}-3,\quad t_{1}=-2$
8. Find a recursive formula for the sequence $16,8,4,2,1,\cdots$.
   
   $t_{n}=\frac{1}{2}\times t_{n+1},\quad t_{1}=16$
   $t_{n}=2\times t_{n-1},\quad t_{1}=16$
   $t_{n}=\frac{1}{2}+t_{n+1},\quad t_{1}=16$
   $t_{n}=\frac{1}{2}\times t_{n-1},\quad t_{1}=16$
9. A recursive formula is given by $t_{n}=\Big(t_{n-1}\Big)^2+1,\quad t_{1}=1$. Find $t_{5}$.
   
   $676$
   $677$
   $36$
   $26$
10. A recursive formula is given by $t_{n}=t_{n+1}+b,\quad t_{1}=1$. If $t_{6}=51$, find $b$
    
    $b=\frac{51}{6}$
    $b=9$
    $b=10$
    $b=11$