Question

Which term of an A.P. $ 7,3, - 1,.... $ is $ - 89 $

Answer 1

Hint: In order to determine position of term $ - 89 $ in the series , first identify the first term of the series and calculate the common difference by subtracting any two consecutive terms. Assume the nth term as $ - 89 $ and put the values of $ a,d,{a_n} $ in the general nth term i.e. $ {a_n} = a + (n - 1)d $ . Solve the equation for n to obtain the required answer.

Complete step-by-step answer:
Clearly, the given sequence is an Arithmetic Progression (A.P.) $ 7,3, - 1,.... $ .
nth term of an A.P. is $ {a_n} = a + (n - 1)d $
where a is the first term, d is the common constant difference
In our sequence first term $ a = 7 $ and
 $ d $ can be calculated by finding the difference of any two consecutive terms of the series $ d = 3 - 7 = - 4 $ .
Let assume the nth term of the A.P be $ {a_n} = - 89 $ ,
Now finding the value of n i.e. position of term $ - 89 $ in the series by putting $ {a_n} = - 89 $ , $ a = 7 $ and $ d = - 4 $ in the general nth term, we have
 $ - 89 = 7 + (n - 1)\left( { - 4} \right) $
Resolving the brackets using distributive law of multiplication as $ A\left( {B + C} \right) = AB + AC $
 $
   - 89 = 7 - 4n + 4 \\
   - 89 - 7 - 4 = - 4n \\
   - 100 = - 4n \\
   - 4n = - 100 \;
  $
Dividing both sides of the equation with the coefficient of $ n $ i.e. $ - 4 $ , we get the value of n as
 $
  \dfrac{{ - 4n}}{{ - 4}} = \dfrac{{ - 100}}{{ - 4}} \\
  n = 25 \;
  $
Therefore, -89 is the 25th term of the given A.P series.
So, the correct answer is “ 25th term”.

Note: 1.Sequence: A sequence is a function whose domains is the set of N of natural numbers.
2.Real Sequence: A sequence whose range is a subset of R is called a real sequence.
In other words, a real sequence is a function having domain N and range equal to a subset of the set R of real numbers.
3.Arithmetic Progression (A.P): A sequence is called an arithmetic progression if the difference of a term and the previous term is always the same.
i.e. $ {a_{n + 1}} - {a_n} = $ constant $ ( = d) $ for all $ n \in N $ .
The constant difference is generally denoted by d which is called as the common difference.