IEB Grade 12 Sequences and Series Cheat Sheet

All the formulas and methods for number patterns, sequences and series (about 25 marks of Paper 1).

Arithmetic (constant difference $d$)

• $T_n=a+(n-1)d$

• $S_n=\frac{n}{2}[2a+(n-1)d]=\frac{n}{2}(a+l)$

• Two terms given: subtract the equations to find $d$.

Geometric (constant ratio $r$)

• $T_n=a{r}^{n-1}$

• $S_n=\frac{a(r^n-1)}{r-1}$, $r\ne 1$

• Two terms given: divide them, e.g. $\frac{T_5}{T_2}=r^3$.

• Unknown $n$ in a power: use logarithms, then decide whether to round up.

Sum to infinity

• Converges only if $-1<r<1$.

• $S_{\infty }=\frac{a}{1-r}$

• Recurring decimals: $0,\dot{3}\dot{6}=\frac{0,36}{1-0,01}=\frac{4}{11}$.

• Bouncing ball: first drop once, every rebound twice (up and down).

Sigma notation

• Number of terms = top $-$ bottom $+1$.

• Write out the first two or three terms to find $a$ and $d$ or $r$. The coefficient is not always $a$.

• Middle terms: $T_{11}+\dots +T_{30}=S_{30}-S_{10}$.

Quadratic patterns

• Constant second difference: $T_n=a{n}^2+bn+c$.

• $2a$ = second difference;
  $3a+b=T_2-T_1$;
  $a+b+c=T_1$.

• First differences form an arithmetic sequence.

Proofs you must know

• Arithmetic: write $S_n$ forwards and backwards and add.

• Geometric: write $S_n$ and $r{S}_n$ and subtract.

Know by heart

• Both sum proofs, line by line

• The convergence condition $-1<r<1$

• The three equations for $a$, $b$ and $c$ in quadratic patterns

On the formula sheet

• $T_n$ and $S_n$ for arithmetic and geometric sequences

• $S_{\infty }=\frac{a}{1-r}$

Common traps

• Using $T_n=a+nd$ or $a{r}^n$.

• Using $S_{30}-S_{11}$ for the sum from $T_{11}$.

• Rounding $n$ down when a sum must exceed a value.

• Using $S_{\infty }$ when $|r|\ge 1$.