Question

Find:

(i) Which term of the A.P. 3, 8, 13, ... is 248?

(ii) Which term of the A.P. 84, 80, 76, ... is 248?

(iii) Which term of the A.P. 4, 9, 14, ... is 254?

(iv) Which term of the A.P. 21, 42, 63, 84, ... is 420?

(v) Which term of the A.P. 121, 117, 113, ... is its first negative term?

(vi) Which term of the A.P. –7, –12, –17, –22,... will be –82? Is –100 any term of the A.P?

Answer 1

In the given problem, we are given an A.P and the value of one of its term.

We need to find which term it is (n)

So here we will find the value of n using the formula,

(i) Here, A.P is

Now,

Common difference (d) =

=

= 5

Thus, using the above mentioned formula

Thus,

Therefore 248 is the of the given A.P

(ii) Here, A.P is

Now,

Common difference (d) =

=80-84

= -4

Thus, using the above mentioned formula

On further simplifying, we get,

Thus,

Therefore 84 is the of the given A.P

(iii) Here, A.P is

Now,

Common difference (d) =

= 9-4

= 5

Thus, using the above mentioned formula

Thus,

Therefore 254 is the of the given A.P

(iv) Here, A.P is

Now,

Common difference (d) =

= 42-21

= 21

Thus, using the above mentioned formula

Thus,

Therefore 420 is the of the given A.P

(v) Here, A.P is

We need to find first negative term of the A.P

Now,

Common difference (d) =

Now, we need to find the first negative term,

Further simplifying, we get,

Thus,

Therefore, the first negative term is the of the given A.P.

(vi) –7, –12, –17, –22,...

$$\begin{gathered} \mathrm{Given}: \\ a=-7 \\ d=-12-\left(-7\right) \\ =-12+7 \\ =-5 \\ {a}_n=-82 \\ \mathrm{Now}, \\ {a}_n=a+\left(n-1\right)d \\ \Rightarrow -82=-7+\left(n-1\right)\left(-5\right) \\ \Rightarrow -82=-7-5n+5 \\ \Rightarrow -82=-2-5n \\ \Rightarrow -82+2=-5n \\ \Rightarrow -80=-5n \\ \Rightarrow n=\frac{-80}{-5} \\ \Rightarrow n=16 \\ \mathrm{Hence},{16}^{\mathrm{th}}\mathrm{term}\mathrm{of}\mathrm{the}\mathrm{A}.\mathrm{P}.\mathrm{is}-82. \\ \mathrm{If}{a}_n=-100 \\ {a}_n=a+\left(n-1\right)d \\ \Rightarrow -100=-7+\left(n-1\right)\left(-5\right) \\ \Rightarrow -100=-7-5n+5 \\ \Rightarrow -100=-2-5n \\ \Rightarrow -100+2=-5n \\ \Rightarrow -98=-5n \\ \Rightarrow n=\frac{-98}{-5} \\ \Rightarrow n=19.6\mathrm{which}\mathrm{is}\mathrm{not}\mathrm{possible}\mathrm{since}n\mathrm{is}\mathrm{a}\mathrm{natural}\mathrm{number} \\ \mathrm{Hence},-100\mathrm{is}\mathrm{not}\mathrm{the}\mathrm{term}\mathrm{of}\mathrm{this}\mathrm{A}.\mathrm{P}. \end{gathered}$$