Question

The Ratio of the sum of ${n^{th}}$ term of $2A.P$’$s$ is $7n + 1:4n + 27$ find the ratio of this ${m^{th}}$ term$?$

Answer 1

Hint: The sum of the first n terms can be calculated for an AP, if the first term and the total terms are known, in the above questions based on progression. An arithmetic sequence as a sequence of numbers in which the second number is obtained for every pair of consecutive terms by adding a fixed number to the first one.

Formula used:
The ${n^{th}}$ term of arithmetic progression formula,
${a_n} = a + (n - 1) \times d$
Where,
$a = $ First term
$d = $ Common difference
$n = $ number of terms
${a_n} = $ nth term
Common difference can be written as, $a,a + 2d,a + 3d,a + 4d,................a + (n - 1) \times d$
For an ${1^{st}}$$A.P$’s ${S_1} = 2{a_1} + (n - 1){d_1}$
For an ${2^{nd}}$ $A.P$’s ${S_2} = 2{a_2} + (n - 1){d_2}$
Where,
${S_1},{S_2}$ is representing a Sum.

Complete step-by-step answer:
Given by,
Ratio of sum of $n$ term of $2A.P$’s$ = (7n + 1):\left( {4n + 27} \right)$
Let as assume the ratio of these $2A.P$’s ${m^{th}}$ terms as $am:a'm$
We know that,
 The ${n^{th}}$ term of arithmetic progression formula,
${a_n} = a + (n - 1) \times d$
$n$ is replaced by $m$
Substituting the given equation becomes,
$\Rightarrow$ $am:a'm = a + (m - 1)d:a'+(m - 1)d'$
On multiplying by $2$ to the right side of the equation,
We get,
$\Rightarrow$ $am:a'm = [2a + 2(m - 1)d]:[2a' + 2(m - 1)d']$
Simplifying the above equation,
Here,
$\Rightarrow$ \[am:a'm = [2a + \{ (2m - 1) - 1\} d]:[2a' + \{ (2m - 1) - 1\} d']\]
Substituting the given formula in above equation,
We know that, for second $A.P$’s
$\Rightarrow$ ${S_1} = 2{a_1} + (n - 1){d_1}$
$\Rightarrow$ ${S_2} = 2{a_2} + (n - 1){d_2}$
$\Rightarrow$ $am:a'm = {S_2}m - 1:{S_2}'m - 1$
Substituting the above formula is applied to the given equation,
We get,
$\Rightarrow$ $am:a'm = $$[7(2m - 1) + 1]:[4(2m - 1) + 27]$
 On Simplifying,
$\Rightarrow$ $am:a'm = $$[14m - 7 + 1]:[8m - 4 + 27]$
$\Rightarrow$ $am:a'm = $ $[14m - 6]:[8m + 23]$
Hence, thus the ratio of the ${m^{th}}$ term of \[2\]$A.P$’s is $[14m - 6]:[8m + 23]$.

Note: A progression is a particular sequence category for which a formula for the nth term can be obtained. The Arithmetic Progression is the most frequently used sequence in arithmetic with formulas that become easy to understand. If the value of common difference d is positive, then the member terms will grow towards positive infinity, and if the value of d is negative, then the member terms will grow towards negative infinity.