Arithmetic case. Spot the COMMON DIFFERENCE d.
unβ=a+(nβ1)d.
Worked example. Sequence 4,11,18,25,β¦. Find unβ.
- a=4, d=7.
- unβ=4+(nβ1)Γ7=4+7nβ7=7nβ3.
- Check: u1β=4. β u2β=11. β
Geometric case. Spot the COMMON RATIO r.
unβ=arnβ1.
Worked example. Sequence 5,15,45,135,β¦. Find unβ.
- a=5, r=3.
- unβ=5Γ3nβ1.
Quadratic case (Extended). If SECOND differences are constant.
Worked example. Sequence 3,7,13,21,31,β¦.
- First differences: 4,6,8,10 (not constant β not AP).
- Second differences: 2,2,2 (constant β QUADRATIC).
- Formula: unβ=an2+bn+c with a=22ndΒ diffβ=1.
- So unβ=n2+bn+c. Use two terms to find b,c.
- n=1: 1+b+c=3βb+c=2.
- n=2: 4+2b+c=7β2b+c=3.
- Subtract: b=1, then c=1.
- Formula: unβ=n2+n+1. Check: u3β=9+3+1=13. β