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After inserting n A.M.’s between 2and 38, the sum of the resulting progressions is 200. The value of n is?
A. 7
B. 8
C. 9
D. 10

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Last updated date: 25th Jun 2024
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Answer
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Hint: Arithmetic mean is the ratio of all observations to the total number of observations. Arithmetic mean represents a number that is obtained by dividing the sum of the elements of a set by the number of values in the set.
Arithmetic Mean = $\dfrac{{{a_1} + {a_2} + {a_3} \ldots + {a_n}}}{n}$
An arithmetic progression (A.P.) is a progression in which the difference between two consecutive terms is constant. If ‘a’ is the first term, ‘l’ is the last term of the A.P and ‘n’ is the number of terms, then the sum of n terms of an A.P. is given by:
${S_n} = \dfrac{n}{2}\left( {a + l} \right)$
Here, after inserting n terms between 2 and 38 then total terms will be n+2. We have to find the value of n by using the formula of sum of A.P as the sum of progression is given.

Complete step by step solution: If there are ‘n’ arithmetic means between 2 and 38, then there are ‘n+2’ terms.
The first term = a = 2
And the last term = l = 38
We know that,
${S_N} = \dfrac{N}{2}\left( {a + l} \right)$
N be the number of terms in an A.P.
Here, N = n+2, a = 2, l = 38 and ${S_{n + 2}} = 200$
$\begin{gathered}
  \therefore {S_{n + 2}} = \dfrac{{n + 2}}{2}\left( {2 + 38} \right) \\
  200 = \dfrac{{n + 2}}{2}\left( {2 + 38} \right) \\
  400 = 40\left( {n + 2} \right) \\
  10 = n + 2 \\
  n = 8 \\
\end{gathered} $
Hence, the value of n is 8
∴Option (B) is correct.

Note: The sum to n terms of an A.P. is also given by:
${S_n} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$
Where d is the common difference i.e. the difference between two consecutive terms in an A.P.
For any A.P., if the common difference is:
1. Positive, the A.P. is increasing
2. Zero, the A.P. is constant
3. Negative, the A.P. is decreasing