Question

Which term of the A.P.: $5\,,\,2\,,\, - 1\,,\,....\,\,is\,\, - 49$.

Answer 1

Hint: Here first we have to find the first term and the common difference of the arithmetic progression. Then we have to apply the formula of ${n^{th}}$ term to find which term is $ - 49$ from the above arithmetic progression.

Complete step by step answer:
In the above question, an A.P. is given as $5\,,\,2\,,\, - 1\,,\,....\,\,is\,\, - 49$.
We know that a general series in arithmetic progression looks like:
$a\,,\,a + d\,,\,a + 2d\,,\,a + 3d\,,\,a + 4d\,...........a + \left( {n - 1} \right)d$ which comprises n terms.
Here, a is the first term and d is the common difference.
Therefore, on comparing the given series with the general series, we get
$a = 5$
Also,
$a + d = 2........\left( 1 \right)$
On substituting the value of a in above equation, we get
$5 + d = 2$
$d = 2 - 5$
$d = - 3$
Now, we will apply the formula of finding the ${n^{th}}$ term of an arithmetic progression.
${a_n} = {a_1} + \left( {n - 1} \right)d$
Here, ${a_n} = - 49\,,\,\,{a_1} = a = 5\,\,,\,\,d = - 3$
Now, substitute the above values in equation
$ - 49 = 5 + \left( {n - 1} \right)\left( { - 3} \right)$
On multiplication in right hand side, we get
$ - 49 = 5 - 3n + 3$
Now, on transposing we get
$3n = 5 + 49 + 3$
$ \Rightarrow 3n = 57$
Now divide both sides by $3$
$ \Rightarrow n = 19$
Therefore, $19^{th}$ term of the given A.P. is $ - 49$.

Note:
An arithmetic progression is a progression or sequence of numbers such that the difference between any two consecutive numbers is constant. For example, the sequence \[5,{\text{ }}9,{\text{ }}13,{\text{ }}17 \ldots .\] is an arithmetic progression or arithmetic sequence whose common difference can be obtained by subtracting one term from its next term, the common difference in this example is $4$.Whenever we face such a problem the key concept is that we have to remember all the formulas of an AP, it will give you a lot of help in finding your desired answer.