Find the explicit formula of the given sequence.

Question

8. arithmetic: a2 = 5 , a5 = -4

 

 

9. geometric: a2 = 75 , a5 = 600

Answer 1

Arithmetic series

Between a2 and a5 we have a3 and a4. If the common difference is d, then a3=a2+d, a4=a3+d, a5=a4+d, so a5-a3=3d, because a4=a3+d=a2+d+d=a2+2d, and a5=a4+d=a2+2d+d=a2+3d. We are given a2=5 and a5=-4. a5-a2=-9=3d, so d=-3. a1=a2+3=8, a0=a1+3=11. Now we can write a formula for a(n)=11-3n, where n starts from 0. Check: a2=11-6=5 and a5=-4.

Geometric series

a5=a2r^3, where r is the common ratio. So 600=75r^3 and r^3=600/75=8, so r=2. a1=a2/2=75/2=37.5 and a0=a1/2=18.75. The formula for a(n)=18.75*2^n. Check: a2=18.75*4=75, a5=18.75*32=600.