Progression Introduction
Arithmetic Progression
- An AP can be represented as a, (a + d), (a + 2d), (a + 3d), ...
Where, a = First term and d is the common difference of the given series.
nth term of an AP
- tn = a + (n - 1)d
Where, tn = nth term, a = First term, d = Common difference and n = Number of terms
Sum of the given number of a series in AP
- Sum of Series = $\cfrac{\text{n}}{2}\ [2\text{a}+(\text{n}-1)\text{d}]$
- Sum of Series = $\cfrac{\text{n}}{2}$ [First term + Last term]
Where, a = First term, d is the common difference of the given series and n = Number of terms
Selecting terms in AP
- 3 terms in AP are a - d, a, a + d
- 4 terms in AP are a - 3d, a - d, a + d, a + 3d and so on.
Geometric Progression
- GP can be represented as a, ar, ar2, ...
Where, a = First term and r is the common ratio of the given series.
nth term of a GP
- tn = arn-1
Where, tn = nth term, a = First term, r = Common ratio and n = Number of terms
Sum of the given number of a series in GP
Sum of Series = $\cfrac{\text{ar}^{\text{n}}-1}{\text{r}-1}\text{ [ if r > 1]}$
Sum of Series = $\cfrac{1-\text{ar}^{\text{n}}}{1-\text{r}}\text{ [ if r < 1]}$
Where, a = First term, r = Common ratio and n = Number of terms
Selecting terms in GP
- 3 terms in GP are a, ar, ar2
- 4 terms in GP are $\cfrac{\text{a}}{\text{r}^3},$ $\cfrac{\text{a}}{\text{r}},$ $\text{ar},$ $\text{ar}^3$ and so on.
The sum of an infinite GP a, ar, ar2, ... is
$\text{S}_\infty =\cfrac{\text{a}}{1-\text{r}}$ $\text{ [ if -1 < r < 1]}$
Where, a = First term and r = Common ratio
Harmonic Progression
If a, b, c are in Arithmetic Progression then 1/a, 1/b, 1/c are in harmonic progression.
nth term of HP
- $ t_{\text{n}}=\cfrac{1}{\text{a}+ (\text{n} -1)\text{d}}$
Where, tn = nth term, a = First term, d = Common difference and n = Number of terms
Selecting terms in HP
- 3 terms in HP are $\cfrac{1}{\text{a}-\text{d}}\text{ ,}$ $ \cfrac{1}{\text{a}} \text{ ,}$ $\cfrac{1}{\text{a}+\text{d}}$
- 4 terms in HP are $\cfrac{1}{\text{a}-3\text{d}} \text{ ,}$ $\cfrac{1}{\text{a}-\text{d}} \text{ ,}$ $\cfrac{1}{\text{a}+\text{d}} \text{ ,}$ $\cfrac{1}{\text{a}+3\text{d}}$ and so on.
If a, b, c are in HP, then $b= \cfrac{2\text{ac}}{\text{a}+\text{c}}$
nth term of HP
- $ t_{\text{n}}=\cfrac{1}{\text{a}+ (\text{n} -1)\text{d}}$
Where, tn = nth term, a = First term, d = Common difference and n = Number of terms
Selecting terms in HP
- 3 terms in HP are $\cfrac{1}{\text{a}-\text{d}}\text{ ,}$ $ \cfrac{1}{\text{a}} \text{ ,}$ $\cfrac{1}{\text{a}+\text{d}}$
- 4 terms in HP are $\cfrac{1}{\text{a}-3\text{d}} \text{ ,}$ $\cfrac{1}{\text{a}-\text{d}} \text{ ,}$ $\cfrac{1}{\text{a}+\text{d}} \text{ ,}$ $\cfrac{1}{\text{a}+3\text{d}}$ and so on.
Relationship between Arithmetic, Harmonic and Geometric Means
- AM = $\cfrac{\text{a}+\text{b}}{2}$
- GM = $\sqrt{\text{ab}}$
- HM = $\cfrac{2\text{ab}}{\text{a}+\text{b}}$
NOTE :
Let there are two numbers a and b where a, b > 0
- AM x HM = GM2
- AM > GM > HM