Question: If 4 times the \[{4^{th}}\] term of an AP is equal to 18 times the \[{18^{th}}\] term, then find the...

Question

If 4 times the ${4^{th}}$ term of an AP is equal to 18 times the ${18^{th}}$ term, then find the ${22^{nd}}$ term.

Answer 1

Solution

Here, we will use the general formula of a series in AP to find the given terms. Then we will use the given condition to form an equation. We will then solve this equation to find the first term of the AP. We will then substitute the value of the first term in the expression for the ${22^{nd}}$ term to get the required answer.

Formula Used:
General term of an AP is: $a + \left( {n - 1} \right)d$ , where $a$ is the first term, $d$ is the common difference and $n$ is the number of the term.

Complete step by step solution:
Let the first term of an Arithmetic Progression (AP) be $a$ the common difference be $d$ .
Now, we know that the general expression of an AP is represented as $a + \left( {n - 1} \right)d$ .
Now, 4 times the ${4^{th}}$ term of an AP is equal to 18 times the ${18^{th}}$ term
${4^{th}}$ term of an AP will be:
${4^{th}}$ term $= a + \left( {4 - 1} \right)d = a + 3d$
Also, ${18^{th}}$ term of an AP will be:
${18^{th}}$ term $= a + \left( {18 - 1} \right)d = a + 17d$
Hence, now converting the given statement into mathematical expression and substituting the above values of the term, we get,
$4\left( {a + 3d} \right) = 18\left( {a + 17d} \right)$
Now, opening the brackets by multiplying the term present outside the bracket by each term inside it, we get,
$\Rightarrow 4a + 12d = 18a + 306d$
$\Rightarrow 12d - 306d = 18a - 4a$
Subtracting the like terms, we get
$\Rightarrow - 294d = 14a$
Dividing both sides by 14, we get
$\Rightarrow a = - 21d$
Now, we are required to find the ${22^{nd}}$ term of this AP.
Again using the general formula, ${22^{nd}}$ term can be written as:
${22^{nd}}$ term $= a + \left( {22 - 1} \right)d = a + 21d$
Now, substituting $a = - 21d$ in the above equation, we get,
${22^{nd}}$ term $= - 21d + 21d = 0$

Therefore, the ${22^{nd}}$ term of this AP is 0.

Note:
An Arithmetic Progression is a sequence of numbers such that the difference between any term and its preceding term is constant. This difference is known as the common difference of the Arithmetic Progression (AP). A real-life example of AP is when we add a fixed amount to our money bank every week. Similarly, when we ride a taxi, we pay an amount for the initial kilometer and pay a fixed amount for all the further kilometers, this also turns out to be an AP.