Sequences and Series
What the domain tests
Sequences and series examines recognition of arithmetic and geometric patterns, nth terms, finite sums and convergence. MSCE Additional Mathematics questions often embed a sequence in an algebraic or practical context, so you must justify the common difference or ratio before applying a formula.
Core rules
- An arithmetic progression has terms a, a+d, a+2d,...; nth term T_n=a+(n-1)d. Sum S_n=n/2[2a+(n-1)d]=n/2(a+l), where l is the last term.
- A geometric progression has terms a, ar, ar^2,...; nth term T_n=ar^{n-1}. Sum S_n=a(1-r^n)/(1-r) for r≠1. An infinite sum exists only if |r|<1, and then S∞=a/(1-r).
- For three consecutive terms of a GP, the square of the middle term equals the product of its neighbours.
- If a sequence is neither arithmetic nor geometric, inspect first and second differences or look for a simple recurrence.
Worked example
The 3rd term of an AP is 11 and the 8th term is 31. Find the sum of the first 20 terms. T_3=a+2d=11 and T_8=a+7d=31. Subtracting gives 5d=20, so d=4. Then a=11-2(4)=3. S_20=20/2[2(3)+19(4)]=10[6+76]=820.
Traps
Using n instead of n-1 in the nth term; applying S∞ when |r|≥1; confusing the denominator 1-r with r-1; failing to state the convergence condition |r|<1.