Sequences, Series and Mathematical Induction
Sequences, Series and Mathematical Induction tests recognition and manipulation of arithmetic and geometric progressions, telescoping sums, and formal induction. You must justify convergence and structure induction clearly.
Core rules
- AP: u_n=a+(n-1)d, S_n=n/2(2a+(n-1)d). GP: u_n=ar^{n-1}, S_n=a(1-r^n)/(1-r) for r≠1; sum to infinity exists only if |r|<1 and equals a/(1-r).
- Method of differences: write each term as f(r)-f(r+1) or similar, cancel interior terms, and keep first and last remnants.
- Induction: prove the base case; assume true for n=k; use the assumption to prove n=k+1; state the conclusion for all n≥base.
Worked example
Sum r/(r+1)! from r=1 to n. Note r/(r+1)! = 1/r! - 1/(r+1)!. Then the sum is (1/1!-1/2!)+(1/2!-1/3!)+...+(1/n!-1/(n+1)!) = 1 - 1/(n+1)!. The interior cancels, leaving only the first positive and final negative terms.
Traps
Starting index mismatch; treating a divergent GP as summable; an induction proof that never uses the assumption; factorial algebra errors; forgetting to state the final conclusion. Also check whether the question asks for a sum, a term, or convergence.