PROGRESSION (A.P., G.P., H.P.) (PART-II)

by Anurag Chandra

Ask The Experts

Properties of
- $\overset {j}{\underset {i=1}{\sum}}2=2+2+\cdots to \quad j \quad terms=2j$
- $\overset {k}{\underset {j=1}{\sum}}2j =2 \overset {k}{\underset {j=1}{\sum}}j$
- $\overset {n}{\underset {k=1}{\sum}}(k^2 + k)=\overset {n}{\underset {k=1}{\sum}}k ^2 + \overset {n}{\underset {k=1}{\sum}}k$
- Odd number of numbers in A.P. whose sum is known may be taken as
  $\cdots \alpha -2\beta, \alpha -\beta, \underline \alpha \quad \alpha +\beta, \alpha +2\beta \cdots$
  go on $\leftarrow \qquad \qquad \qquad \qquad \qquad \rightarrow$ go on
  $subtracting \quad \beta \qquad \qquad \qquad adding \quad \beta$
- Even number of numbers in A.P. whose sum is known may be taken as
  $\cdots \alpha -5\beta, \alpha-3\beta, \underline{\alpha -\beta, \alpha +\beta}, \alpha +3\beta, \alpha +5\beta,\cdots$
  
  go on $\leftarrow \qquad \qquad \qquad \qquad \qquad \rightarrow$ go on
  $subtracting \quad 2\beta \qquad \qquad \qquad adding \quad 2\beta$
- When sum is given:
  1. Three numbers in A.P. may be taken as $\alpha -\beta, \alpha, \alpha +\beta$
  2. Four numbers in A.P. may be taken as $\alpha -3\beta, \alpha -\beta, a + \beta, \alpha +3\beta$
  3. Five numbers in A.P. may be taken as $\alpha -2\beta, \alpha -\beta, \alpha, \alpha +\beta, \alpha +2\beta$
- Odd number of numbers in G.P. whose product is known may be taken as $\cdots \cdots ,\frac {a}{r^2}.\frac {a}{r}, \underline a, ar, ar^2,\cdots$
  go on $\leftarrow \qquad \qquad \qquad \qquad \qquad \rightarrow$ go on
  $dividing \quad by \quad r \qquad \qquad \qquad multiplying \quad by \quad r$
- Even number of numbers in G.P. whose product is known may be taken as
  $\cdots ,\frac {a}{r^5}, \frac {a}{r^3}, \frac {a}{r}, ar, ar^3, ar^5 \cdots$
  
  go on $\leftarrow \qquad \qquad \qquad \qquad \rightarrow$ go on
  $dividing \quad by \quad r^2 \qquad \qquad multiplying \quad by \quad r^2$
- When product is given:
  1. Three numbers in G.P. may be taken as $\frac {a}{r}, a, ar$
  2. Four numbers in G.P. may be taken as $\frac {a}{r^3}, \frac {a}{r}, ar, ar^3$
  3. Five numbers in G.P. may be taken as $\frac{a}{r^2},\frac{a}{r},a,ar,ar^2$
- For A.P. :
  1. $a,a+d,a+2d, \cdots ,b-2d,b-d,b$ go on adding $d \leftarrow \quad \rightarrow$ go on subtracting d is an A.P. whose first term is a, last term is b and common difference is d.
  2. Sum of equidistant terms from beginning and end = constant = a + b
    = first term + last term
  3. If the same number is added to or subtracted from all the terms of an A.P., then the resulting sequence is also an A.P. with the same common difference as before.
  4. If all the terms of an A.P. with common difference d be multiplied (or divided) by the same number k, then the resulting sequence is also an A.P. with common difference $kd\Big( or \frac {d}{k} \Big)$
- 1. A sequence is an A.P. iff its nth term is of the form an + b, where a, b are constants. The common difference of this A.P. is a (coefficient of n.)
  2. A sequence is an A.P. iff the sum of its first n terms is of the form , where a, b, c are constants. The common difference of this A.P. is 2a (two times the coefficient of )
- For G.P. :
  1. $a,ar,ar^2,\cdots \frac {b}{r^2}, \frac {b}{r}, b$
    go on multiplying by r $\leftarrow \qquad \rightarrow$ go on dividing by r
    is a G.P. whose first term is a, last term is b and common ratio is r
  2. The product of equidistant terms from the beginning and the end
    =constant=ab
    =first terms * last term
Sum of different types of series:
If all the terms of a G.P. with common ratio r be multiplied (or divided) by the same number k then the resulting sequence is also a G.P. having common ratio
1. When difference of terms of a series are in A.P. or G.P. $t_n(n^{th} term)=t_1+[sum \quad to (n-1)$ terms of the series formed by difference of terms]
  and $S_n=\overset {n}{\underset {n=1}{\sum}}t_n$
2. To find the sum of n terms of a series whose $n^{th}$ term
  $t_n= \frac {1}{(an+b)(cn+d)(pn+q)}=f(n)say$
  Write down $t_n = \frac {value \quad of (an+b)f(n)at \quad n = - b/a} {an+b}+\frac {value \quad of (cn+d)f(n)at \quad n=- \frac {d}{c}}{cn+d}$
  $+ \frac {value \quad of (pn+q)f(n)at \quad n = - \frac{q}{p}}{pn+q}$
  Put $n=1,2,3,\cdots ,n$ and add.
3. Sum of arithmetico-geometric series
  Let $a+(a+d)r+(a+2d)r^2 +\cdots +[a+(n-1)d]r^{n-1}$ be the given arithmetico-geometric series
  Let $S=a+(a+d)r+(a+2d)r^2 + \cdots + [a+(n-1)d]r^{n-1} \qquad \cdots (1)$
  then $rS=ar+(a+d)r^2+ \cdots +[a+(n-2)d]r^{n-1}+[a+(n-1)d]r^n \qquad \cdots (2)$
  Subtracting (2) from (1)
  $S=\frac {a}{1-r}+ \Big( r-\frac {r^n}{1-r} \Big) d- \frac {[a+(n-1)d]r^n}{1-r}$
  where $r \not = 1$
- If an A.P. and an H.P. have the same first term a, the same last term b and the same number of terms, then any term ( $r ^{th}$ term) of A.P. from beginning Ã- corresponding term ( $r ^{th}$ term) of H.P. from end
  
  $=\Big( \frac {an+(r-1)b-ra}{n-1}\Big) .\Big( \frac {(n-1)ab}{na+(r-1)b-ra}\Big) =first \quad term * last \quad term=ab$
  
  Explanation
  Required $\quad$ product $=\Big[a+(r-1)\frac{b-a}{n-1}\Big]$ . $\frac {1}{\frac {1}{b}-(r-1)\frac {a-b}{(n-1)ab}}=\Big(\frac {an-a+(r-1)b-ra+a}{n-1}\Big)$
  
  $\Big[\frac {(n-1)ab}{na-a-ra+a+(r-1)b}\Big]$
- If from three numbers in H.P., half the middle term is subtracted, the resulting numbers are in G.P.
  Thus if a, b, c are in H.P., then
  
  $a- \frac {b}{2},\frac {b}{2},c- \frac {b}{2}$ are in H.P.
  
  Explanation:
  $\Bigg(a-\frac {b}{2}\Bigg)$ $\Bigg(c-\frac {b}{2}\Bigg)=$ $\Bigg(a-\frac{ac}{a+c}\Bigg)$ $\Bigg(c-\frac {ac}{a+c}\Bigg)=$ $\frac{a^2}{a+c}.\frac {c^2}{a+c}=\Bigg( \frac {ac}{a+c}\Bigg)^2 =\Bigg( \frac {b}{2}\Bigg)^2$
- If $a_1,a_2,a_3, \cdots ,a_n$ are in H.P., then
  $a_1a_2+a_2a_3+\cdots \cdots +a_{n-1}a_n=(n-1)a_1a_n$
  
  Explanation:
  $a_1a_2=\frac {1}{d}(a_1-a_2) \qquad \Big[ \therefore \frac {1}{a^2}-\frac {1}{a_1}=d \Big]$
  $a_2a_3=\frac {1}{d}(a_2-a_3)$
  $a_{n-1}a_n=\frac {1}{d}(a_{n-1}-a_n)$
  Add and put $d=\frac {a_1-a_n}{(n-1)a_1a_n}$
Inequality:
- A.M., G.M., and H.M. of n positive numbers $a_1,a_2,a_3,\cdots ,a_n$ are given by
  $A=\frac {a_1+a_2+\cdots +a_n}{n}, G=(a_1a_2\cdots a_n)^{1/n}$ , $H=\frac {n}{\frac {1}{a_1}+\frac {1}{a_2}+\cdots +\frac {1}{a_n}}$
- , equality holds only when
  This result should be used when
  - All the numbers involved are positive.
  - Inequality occurs or maximum or minimum value is to be involved
  - Any two of sum of numbers, product of numbers and sum of reciprocals are to be involved.