Sequences and Series
Sequences and Series
A sequence is an ordered list; a series is the sum of its terms. This domain covers arithmetic and geometric patterns, notation, and infinite sums.
Concepts. An arithmetic sequence has constant difference $d$: $t_n=a+(n-1)d$ and $S_n=\dfrac{n}{2}[2a+(n-1)d]=\dfrac{n}{2}(a+t_n)$. A geometric sequence has constant ratio $r$: $t_n=ar^{,n-1}$ and $S_n=\dfrac{a(r^n-1)}{r-1}$ for $r\ne1$. An infinite geometric series converges only when $|r|<1$, with sum $S_\infty=\dfrac{a}{1-r}$. Sigma notation compresses sums, and recursive definitions express each term from earlier ones. Applications include simple and compound interest, depreciation, and annuity-style accumulation.
Traps. The most common slip is confusing $n$, the term number, with the value of the term. "Find the 20th term" means substituting $n=20$, not $t_n=20$. For geometric series, students misapply the formula when $r=1$ (where the sum is simply $na$) or forget that convergence requires $|r|<1$; a common ratio of $1.2$ gives a divergent series no matter how the question is phrased. In application problems, decide whether the first term corresponds to year 0 or year 1 before writing the model. Also watch for sequences that are neither arithmetic nor geometric and must be handled by pattern analysis or recursion.
Rule of thumb. Subtract consecutive terms to test for arithmetic; divide consecutive terms to test for geometric. Whichever gives a constant value identifies the type.
Worked example. A geometric series has $a=6$ and $r=\tfrac13$. Then $t_5=6\left(\tfrac13\right)^4=\tfrac{6}{81}=\tfrac{2}{27}$, and since $|r|<1$ the infinite sum is $S_\infty=\dfrac{6}{1-\tfrac13}=9$.