Question

Continue the two sequences of numbers below and find an equation to each of the sequences: n 1 2 3 4 5 6 7 Equation an 2 5 9 14 20 27 bn 1 3 12 60 360 2520

Please tell the complete solution of this question.

Answer 1

1.

$$a_1=2, a_2=5=a_1+3, a_3=9=a_2+4,$$

$$a_4=14=a_3+5,a_5=20=a_4+6,$$

$$a_6=27=a_5+7$$

$$a_n=a_{n-1}+n+1, n>1$$

$$a_n=\displaystyle\sum_{k=1}^n(k+1)=\dfrac{n(n+1)}{2}+n=\dfrac{n(n+3)}{2}, n\geq1$$

2.

$$b_1=1, b_2=3=b_1(3), b_3=12=b_2(4),$$

$$b_4=60=b_3(5),b_5=360=b_4(6),$$

$$b_6=2520=b_5(7)$$

$$b_n=b_{n-1}(n+1), n>1$$

$$b_n=\dfrac{(n+1)!}{2}, n\geq1$$