Answer

Verified

447.6k+ views

**Hint:**In this problem, we will be using the concept of the sum of an arithmetic progression (A.P). In order to solve this question, we divide the sum of n terms of the two arithmetic progression(A.P) with each other and equate it with $5n+4:9n+6$.

Now we will try to get the equation in the form of the ${\text{n}}^{\text{th}}$ term of an A.P and will find the values for ‘n’. Upon putting that value of ‘n’ in $5n+4:9n+6$. We can calculate the ratio of the ${18}^{\text{th}}$ term of two given A.P.

**Complete step by step solution:**As mentioned in the question, there are two arithmetic progressions with different first terms and different common differences.

For the first A⋅P:-

Let the first term of A⋅P be = a, and the common difference be = d;

So, Sum of ‘n’ terms of an A⋅P is;

${{S}_{n}}=\dfrac{n}{2}\left[ 2a+\left( n-1 \right)d \right]$

and the nth term of an A⋅P is;

${{a}_{n}}=a+\left( n-1 \right)d$

For the second A⋅P:-

Let the first term of A⋅P. be = A and the common difference be = D;

So, the sum of its ‘n’ terms will be;

${{S}_{n}}=\dfrac{n}{2}\left[ 2A+\left( n-1 \right)D \right]$

And the nth term of an A⋅P is;

${{A}_{n}}=A+\left( n-1 \right)D$

It’s given in the question that the ratio of the sum of ‘n’ terms of the two AP is $5n+4:\ 9n+6;$

$\Rightarrow \dfrac{\dfrac{n}{2}\left[ 2a+\left( n-1 \right)d \right]}{\dfrac{n}{2}\left[ 2\text{A}+\left( n-1 \right)\text{D} \right]}=\dfrac{5n+4}{9n+6}$

$\Rightarrow \dfrac{\left[ 2a+\left( n-1 \right)d \right]}{\left[ 2A+\left( n-1 \right)d \right]}=\dfrac{5n+4}{9n+6}$

Taking L.H.S;

When we take ‘2’ common in numerator and denominator;

\[=\dfrac{2\left[ a+\left( \dfrac{n-1}{2} \right)d \right]}{2\left[ A+\left( \dfrac{n-1}{2} \right)D \right]}\]

$=\dfrac{\left[ a+\left( \dfrac{n-1}{2} \right)d \right]}{\left[ A+\left( \dfrac{n-1}{2} \right)D \right]}$

So, now;

\[\dfrac{\left[ a+\left( \dfrac{n-1}{2} \right)d \right]}{\left[ A+\left( \dfrac{n-1}{2} \right)D \right]}=\dfrac{5n+4}{9n+6}\] (1)

We need to find the ratio of the 18th term of Arithmetic progression:

\[=\dfrac{18\text{th}\ \text{term}\ \text{of}\ \text{1st}\ \text{A}\cdot \text{P}}{18\text{th}\ \text{term}\ \text{of}\ \text{2nd}\ \text{A}\cdot \text{P}}\]

$\dfrac{{{a}_{18}}\ \text{of}\ 1\text{st}\ \text{A}\cdot \text{P}}{{{A}_{18}}\ \text{of}\ 2\text{nd}\ \text{A}\cdot \text{P}}$

$=\dfrac{a+\left( 18-1 \right)d}{A+\left( 18-1 \right)D}$

$=\dfrac{a+17d}{A+17D}$ (2)

Comparing equation (2) with equation (1); $a+17d=a+\left( \dfrac{n-1}{2} \right)d$

$\Rightarrow 17=\dfrac{n-1}{2}$

$\Rightarrow n-1=17\times 2$

$\Rightarrow n-1=34$

$\Rightarrow n=34+1$

$\Rightarrow n=35$

Now, putting $n=35$ in equation (1);

\[\Rightarrow \dfrac{\left[ a+\left( \dfrac{n-1}{2} \right)d \right]}{\left[ A+\left( \dfrac{n-1}{2} \right)D \right]}=\dfrac{5n+4}{9n+6}\]

$\Rightarrow \dfrac{a+\left( \dfrac{35-1}{2} \right)d}{A+\left( \dfrac{35-1}{2} \right)D}=\dfrac{5\left( 35 \right)+4}{9\left( 35 \right)+6}$

$\Rightarrow \dfrac{a+\left( \dfrac{34}{2} \right)d}{A+\left( \dfrac{34}{2} \right)D}=\dfrac{175+4}{315+6}$

$\Rightarrow \dfrac{a+17d}{A+17D}=\dfrac{179}{321}$

Therefore, $\dfrac{18\text{th}\ \text{term}\ \text{of}\ 1\text{st}\ \text{A}\cdot \text{P}}{18\text{th}\ \text{term}\ \text{of}\ 2\text{nd}\ \text{A}\cdot \text{P}}=\dfrac{179}{321}$

Hence, the ratio of ${18}^{\text{th}}$ term of ${1}^{\text{st}}$ A⋅P and ${18}^{\text{th}}$ term of ${2}^{\text{nd}}$ A⋅P is 179: 321.

**Note:**The sum of the ‘n’ terms of any A⋅P is ${{S}_{n}}=\dfrac{n}{2}\left[ 2a+\left( n-1 \right)d \right]$ and ${n}^{\text{th}}$ term of an A⋅P is ${{a}_{n}}=a+\left( n-1 \right)d$ where ‘a’ is the first term of an A.P and ‘d’ is common difference of an A.P.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Choose the antonym of the word given below Furious class 9 english CBSE

How do you graph the function fx 4x class 9 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Change the following sentences into negative and interrogative class 10 english CBSE