Question

Is – 150 a term of AP: 11, 8, 5, 2?
A) No
B) Yes
C) Data Insufficient
D) Can’t say

Answer 1

Hint: We can let – 150 as the $ {n^{th}} $ term of the given AP. By using the formula for the same, we can find the position of this term. If the position is an integer then it will be a part of AP and if decimal, then it will not be a part of it.
Formula to be used:
 $ {a_n} = a + \left( {n - 1} \right)d $ where, $ {a_n} $ is the $ {n^{th}} $ term, a is the first term, d is the common difference and n is the position of the $ {n^{th}} $ term

Complete step-by-step answer:
Arithmetic Progression (AP) refers to a sequence of numbers where the difference of any 2 consecutive numbers is the same throughout. This difference is known as a common difference.
The given AP is 11, 8, 5, 2. We can let – 150 as a term of this AP and find its position. If the value of its position comes out to be an integer, then it will be a part of this AP belonging to that calculated position.
Let – 150 be the $ {n^{th}} $ term of the AP is 11, 8, 5, 2.
The formula for $ {n^{th}} $ term of an AP is given as:
 $ {a_n} = a + \left( {n - 1} \right)d $ here,
 $ {n^{th}} $ term $ \left( {{a_n}} \right) $ = - 150
First term (a) = 11
Common difference (d) = - 3 $ \left(
  \because 8 - 11 = - 3,
  5 - 8 = - 3 \;
  \right) $
Position of the $ {n^{th}} $ term = n
Substituting the values and finding the value of n:
 $
  \Rightarrow - 150 = 11 + \left( {n - 1} \right) (- 3) \\
  \Rightarrow - 3\left( {n - 1} \right) = - 150 - 11 \\
  \Rightarrow - 3\left( {n - 1} \right) = - 161 \\
  \Rightarrow - 3n + 3 = - 161 \\
  \Rightarrow - 3n = - 161 - 3 \\
  \Rightarrow - 3n = - 164 \\
   \Rightarrow n = \dfrac{{ - 164}}{{ - 3}} \\
   \Rightarrow n = 54.66 \;
  $
The value of n obtained is a decimal number and not an integer showing that there can be no possible position for this $ {n^{th}} $ term.
Therefore, – 150 is not a term of the given AP and the correct option is A).
So, the correct answer is “Option a”.

Note: Integers are the numbers that include both negative and positive values but fractions and decimals are not counted as integers.
There are certain quantities whose representation is not valid in decimals. Such quantities are population, place of any number in sequence etc.
To find the common difference of AP we can take any 2 consecutive values from the sequence and then subtract the latter from the former.