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In the series $20,18,16,..............., - 2$ is the term:
A) $10^{th}$
B) $11^{th}$
C) $12^{th}$
D) $13^{th}$

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Last updated date: 14th Jun 2024
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Answer
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Hint: According to given in the question we have to determine the number of term -2 for the given series $20,18,16,...............,$So, first of all we have to determine that the given series is in A.P or not and for that we have to determine the common difference which is explained below:
Common difference: For an A.P series common difference can be obtained by finding the difference between the second and first term and same as third and second term and if the difference obtained between second and first terms and third and second terms are same then we can say that given series is in A.P.
Now, to find the number of term -2 we have to use the formula as mentioned below:
$ \Rightarrow {a_n} = a + (n - 1)d................(A)$
Where, a is the first term of the given A.P series, d is the common difference and n is the number of terms which we have to find.

Complete step-by-step solution:
Step 1: First of all we have to check if the given series is in A.P or not which we can check by finding the common difference as mentioned in the solution hint. So, first of all we will find the difference between second and first term. Hence,
First term a = 20, and second term is = 18
Common difference (d): $18 - 20 = - 2$
Step 2: Now, same as the step 1 we have to determine the common difference for third and second terms. Hence,
Second term is = 18, and third term is = 16
Common difference (d): $16 - 18 = - 2$
Step 3: Now, from step 1 and step 2 we obtained the common difference d and we can say that the given series is in A.P.
Step 4: Now, to find the number of terms we have to use the formula (A) as mentioned in the solution hint. Hence, on substituting all the values in the formula (A),
$
   \Rightarrow - 2 = 20 + (n - 1)( - 2) \\
   \Rightarrow - 2 = 20 - 2n + 2
 $
Now, on solving the expression obtained just above,
$
   \Rightarrow - 2 = 22 - 2n \\
   \Rightarrow n = \dfrac{{24}}{2} \\
   \Rightarrow n = 12
 $
Final solution: Hence, with the help of the formula (A) as mentioned in the solution hint we have obtained that -2 is the $12^{th}$.

Therefore option (C) is correct.

Note: It is necessary to check that the given series is an A.P series or not which can be determined by finding the difference between the second and first term and same as third and second term and if the difference obtained between second and first terms and third and second terms are same then we can say that given series is in A.P.