Arithmetic and Geometric Progressions, Series
Arithmetic and Geometric Progressions, Series tests recognition, formula selection and summation. You must distinguish AP from GP, handle finite and infinite sums correctly, and use means and series results without misplacing n.
Core rules
- AP: aₙ=a+(n−1)d; Sₙ=n/2[2a+(n−1)d]=n/2(a+l).
- GP: aₙ=ar^{n−1}; Sₙ=a(rⁿ−1)/(r−1) for r≠1; S∞=a/(1−r) only when |r|<1.
- Arithmetic mean of a and b is (a+b)/2; geometric mean is √(ab) for a,b≥0.
- Sum of first n natural numbers is n(n+1)/2; squares n(n+1)(2n+1)/6; cubes [n(n+1)/2]².
- Three terms in AP: a−d, a, a+d; in GP: a/r, a, ar.
Worked example
The 5th term of an AP is 17 and the 12th is 38. Then 7d=21, so d=3. From a+4(3)=17, a=5. S₂₀=20/2[2(5)+19(3)]=10(10+57)=670.
Traps
Use n−1, not n, in the nth term. The infinite GP formula fails at |r|≥1 and at r=1. Do not use S∞ for a finite series. If a question says terms between, the number of inserted terms is not the total number of terms. Check whether the question asks for a term, a sum, or a mean.