Sequences and Series
Sequences and series questions test recognition, formulas, and convergence. Bagrut problems include arithmetic and geometric sequences, sums, recursive definitions, and infinite geometric series, sometimes embedded in a word problem about growth or savings.
Core rules
- Arithmetic: a_n=a_1+(n−1)d and S_n=n/2·(2a_1+(n−1)d)=n/2·(a_1+a_n).
- Geometric: a_n=a_1·r^(n−1) and S_n=a_1(1−r^n)/(1−r) for r≠1.
- An infinite geometric series converges only when |r|<1, and then S=a_1/(1−r).
- A recursive rule gives the next term from previous terms; you still need initial terms.
- Sigma notation compresses a sum: Σ from k=1 to n of a_k.
Worked example
An arithmetic sequence has a_3=11 and a_8=31. Find S_20. The difference between terms is 5 steps: a_8−a_3=20, so 5d=20 and d=4. Then a_1=a_3−2d=11−8=3. The sum is S_20=20/2·(2·3+19·4)=10·(6+76)=820. The key is counting intervals, not terms: from a_3 to a_8 there are five common differences.
Traps
Using n instead of n−1 in the nth-term formula shifts every answer. For an infinite geometric series, |r|<1 is a condition, not a formality; r=2 diverges. Watch negative ratios: signs alternate, but the sum formula still needs the correct power of r. Do not confuse a_n with S_n.