Sequences and Series

The infinite! No other question has ever moved so profoundly the spirit of man. — David Hilbert

8.1 Sequences and Series

A sequence is a function whose domain is the set of natural numbers (or a finite initial segment). We write $a_1,a_2,a_3,\ldots$

A series is the indicated sum $a_1+a_2+\cdots$

8.2 Arithmetic Progression (A.P.)

$a,\; a+d,\; a+2d,\;\ldots$ with first term $a$ and common difference $d$.

$n\text{th term:}\quad a_n=a+(n-1)d$

$S_n=\frac{n}{2}\bigl(2a+(n-1)d\bigr)=\frac{n}{2}(a+\ell)$

where $\ell$ is the last term.

Arithmetic mean of $a$ and $b$ is $\dfrac{a+b}{2}$.

8.3 Geometric Progression (G.P.)

$a,\; ar,\; ar^2,\;\ldots$ with first term $a$ and common ratio $r\mathrel{\htmlClass{course-neq}{\char"2260}} 0$.

$a_n=ar^{n-1}$

$S_n=a\frac{r^n-1}{r-1}\quad (r\mathrel{\htmlClass{course-neq}{\char"2260}} 1),\qquad S_n=na\quad (r=1)$

If $|r|<1$, the infinite GP sums to $S=\dfrac{a}{1-r}$.

Geometric mean of positive $a,b$ is $\sqrt{ab}$. For positive numbers, $\operatorname{AM}\ge\operatorname{GM}$.

Chapter Summary

• Sequence vs series
• AP: $a_n=a+(n-1)d$, $S_n=\frac{n}{2}(2a+(n-1)d)$
• GP: $a_n=ar^{n-1}$, $S_n=a(r^n-1)/(r-1)$
• Infinite GP: $a/(1-r)$ for $|r|<1$

Exercises (NCERT)

• Exercise 8.1 onwards — sequences, AP, GP
• Miscellaneous Exercise