Question

Find the number of terms in the A.P, 100, 96, 92………….12

Answer 1

Hint: Total number of terms of any sequence is the number of counting’s from first to last number i.e. position of last number is the total number of terms of any sequence. General term of an A.P is given by the relation
${{T}_{n}}=a+\left( n-1 \right)d$
Where ‘a’ is the first term of A.P and ‘d’ is the common difference and ${{T}_{n}}$ is the nth term of it.

Complete step by step answer:
We have sequence of arithmetic progression in the problem as
100, 96, 92…………12……………….(i)
Here, we need to determine the number of terms in the given sequence. As we can observe the sequence and get that the last term of the sequence is 12 and hence the counting from the starting number 100 to the last number 12 will give the total number of terms in the given sequence. First term of the given sequence is 100 and the successive difference or common difference of the A.P can be given as
96 – 100 = 92 – 96 = - 4.
Let us suppose 12 is the nth term of the sequence i.e. there will be n terms of the given sequence as well as 12 is the last term of the A.P. As we know any term of an A.P can be calculated by the formula of general term of an A.P which can be given as
${{T}_{n}}=a+\left( n-1 \right)d...........\left( ii \right)$
Where ${{T}_{n}}$ is the nth term, a is the first term and ‘d’ is the common difference of any given A.P. So we can put values of $'{{T}_{n}}'$ , ‘a’ and ‘d’ in the given equation (ii) from the sequence given in equation (i). So, we have
${{T}_{n}}=12=$ last term (nth term)
a = 100 = first term
d = - 4 = common difference.
So, we can get value of ‘n’ i.e. total number of terms by putting the above values in the equation (ii) we get
12 = 100 + (n – 1) (-4)
12 – 100 = (n – 1) (-4)
-88 = (n – 1) (-4)
Divide the whole equation (-4) to both sides. So, we get
$\dfrac{-88}{-4}=\left( n-1 \right)$
12 = n – 1
n = 13
Hence, 12 is the 13th term of the given sequence of A.P. As 12 is the last term so there will be a total 13 terms of the given sequence in the problem.

Note: One may calculate the number of terms by writing the whole sequence as there are only 13 terms in the sequence. So, you can use this approach here but for the series with a higher number of terms, we cannot apply the above method. So, try to use only identities with these kinds of questions.
Don’t use ${{s}_{n}}=\dfrac{n}{2}\left[ 2a+\left( n-1 \right)d \right]$ as one may get confuse with these two identities used in arithmetic progression chapter. It will give the sum of the given sequence, not the numbers of terms. So, be clear with the identities and don’t confuse them.