Question

The sum of ${5^{th}}$ and ${9^{th}}$ terms of an A.P. is $30$. If its ${25^{th}}$ term is three times its ${8^{th}}$ term, find the A.P.

Answer 1

Here we are asked to find the arithmetic progression using the given data. A sequence of numbers in which the difference of any two consecutive terms is constant is known as arithmetic progression. In general, the ${n^{th}}$ term in the arithmetic progression is written as $a + (n - 1)d$ where $a -$the first term in arithmetic progression, $d -$the common difference between consecutive terms in arithmetic progression. Using this general term, we will find the A.P.

Complete step by step answer:
It is given that the sum of ${5^{th}}$ and ${9^{th}}$ terms of an A.P. is $30$ and the ${25^{th}}$ term is three times its ${8^{th}}$ term. We aim to find the arithmetic progression.
We know that A.P is nothing but the sequence of numbers where the difference of any consecutive term is the same. Also, in general, the ${n^{th}}$ term in the arithmetic progression is written as $a + (n - 1)d$ where $a -$the first term in arithmetic progression, $d -$the common difference between consecutive terms in the arithmetic progression.
From the question, we have that the sum of ${5^{th}}$ and ${9^{th}}$ terms of an A.P. is $30$. Let us first write the ${5^{th}}$ and ${9^{th}}$ terms of an A.P. using the general form.
${5^{th}}$ term - $a + (5 - 1)d = a + 4d$
${9^{th}}$ term - $a + (9 - 1)d = a + 8d$
Therefore, we get $a + 4d + a + 8d = 30$
Let us simplify the above equation.
$\Rightarrow 2a + 12d = 30$
Dividing the above equation by two we get
$\Rightarrow a + 6d = 15$
Re-arranging the above equation we get
$\Rightarrow a = 15 - 6d$
Now we have an expression for the term $a$, let’s keep it for future use. We have also given that the ${25^{th}}$ term is three times its ${8^{th}}$ term. Let us write these terms using the general form
${25^{th}}$ term - $a + (25 - 1)d = a + 24d$
${8^{th}}$ term - $a + (8 - 1)d = a + 7d$
Therefore, we get $a + 24d = 3\left( {a + 7d} \right)$
On simplifying the above equation, we get
$\Rightarrow a + 24d = 3a + 21d$
$\Rightarrow 24d - 21d = 3a - a$
$\Rightarrow 3d = 2a$
Now let us substitute the expression that we found for the term $a$.
$\Rightarrow 3d = 2(15 - 6d)$
Let us simplify the above equation further.
$\Rightarrow 3d = 30 - 12d$
$\Rightarrow 3d + 12d = 30$
$\Rightarrow 15d = 30$
$\Rightarrow d = \dfrac{{30}}{{15}}$
$\Rightarrow d = 2$
Thus, we got the value of the common difference. Substituting this in $a = 15 - 6d$ we get
$\Rightarrow a = 15 - 6(2)$
$\Rightarrow a = 15 - 12$
$\Rightarrow a = 3$
Now we also got the first term in the arithmetic progression. Let us find the arithmetic progression.
Since the first term in A.P. is three and the common difference is two the term sin the A.P. will be $3,5,7,9,11,13,15,...$.
Thus, the required arithmetic progression is $3,5,7,9,11,13,15,...$

Note:
In mathematics, there are three types of progression. They are Arithmetic, Geometric, and Harmonic. The arithmetic progression is denoted as A.P. in short. A progression is nothing but a special type of sequence of numbers and it is also possible to find the formula for the ${n^{th}}$ term.