Question

Which term of the AP: $21,18,15,....$ is -81?

Answer 1

We use the general equation of a term in an arithmetic progression and find the value of ‘n’ after substituting the given values in the formula.

• An arithmetic progression is a sequence of terms having common differences between them. If ‘a’ is the first term of an AP, ‘d’ is the common difference, then the $n^{th}$ term of an AP can be found as ${a_n} = a + (n - 1)d$ .

Complete step-by-step solution:
We are given an AP $21,18,15,....$
We have the first term of the AP as 21
$\Rightarrow a = 21$
Since common difference is the difference between two consecutive terms of an AP
$\Rightarrow d = 18 - 21$
$\Rightarrow d = - 3$
(Verify common difference $d = 15 - 18 = - 3$)
Now we take the value of the nth term as -81
Since we know then nth term of an AP can be found as${a_n} = a + (n - 1)d$ .
Substitute the value of ${a_n} = - 81,a = 21,d = - 3$in the formula
$\Rightarrow - 81 = 21 + (n - 1)( - 3)$
Shift all constant terms to left hand side of the equation
$\Rightarrow - 81 - 21 = (n - 1)( - 3)$
Calculate the value in left hand side of the equation
$\Rightarrow - 102 = (n - 1)( - 3)$
Divide both sides of the equation by -3
$\Rightarrow \dfrac{{ - 102}}{{ - 3}} = \dfrac{{(n - 1)( - 3)}}{{ - 3}}$
Cancel same factors from numerator and denominator on both sides of the equation
$\Rightarrow 34 = n - 1$
Shift all constant terms to left hand side of the equation
$\Rightarrow 34 + 1 = n$
Calculate the value on left hand side of the equation
$\Rightarrow 35 = n$

$\therefore$-81 is the $35^{th}$ term in the AP

Note: Many students make mistakes when shifting values from one side of the equation to another, keep in mind we always change sign from positive to negative and vice-versa when shifting values to the opposite side of the equation. Also, verify the value of common difference by calculating the difference between two pairs of terms of AP.