Question

What is mean of the series \[a\], \[a + nd\], \[a + 2nd\]?
A. \[a + \left( {n - 1} \right)d\]
B. \[a + nd\]
C. \[a + \left( {n + 1} \right)d\]
D. None of these

Answer 1

Hint: First we will add each element of the series. Then we will divide the sum by the number of terms of the series to get the required result.

Formula used
Mean of the series is \[ = \dfrac{{{\rm{Sum}}\,{\rm{of}}\,{\rm{terms}}}}{{{\rm{Number}}\,{\rm{of}}\,{\rm{terms}}}}\]

Complete step by step solution:
Given series is \[a\], \[a + nd\], \[a + 2nd\].
The number of terms of the series is 3.
The sum of terms of the series is \[a + \left( {a + nd} \right) + \left( {a + 2nd} \right)\]
\[ = a + a + nd + a + 2nd\]
Add the like terms
\[ = \left( {a + a + a} \right) + \left( {nd + 2nd} \right)\]
\[ = 3a + 3nd\]
Take common 3
\[ = 3\left( {a + nd} \right)\]
Finding the mean by using the formula of mean
Mean \[ = \dfrac{{3\left( {a + nd} \right)}}{{\rm{3}}}\]
Cancel out 3 from denominator and numerator
\[ = a + nd\]
Hence option B is the correct option.
Additional information:
There are three types of mean.
1) Arithmetic mean: Arithmetic mean is the same as average. It is calculated by adding the all observations divided by the number of observations.
2) Geometric mean: It is calculated by the nth square root of the product of n terms
3) Harmonic mean: The number of terms divided by the sum of the terms.
Note: \[a + nd - a = a+ 2nd - (a+nd) = nd \].So the given series is in Ap. Then the mean of an AP series is \[{\left( {\dfrac{{n + 1}}{2}} \right)^{th}}\] term where n is an odd number. Here the number of terms is 3. So, mean is \[{\left( {\dfrac{{3 + 1}}{2}} \right)^{th}} = {2^{nd}}\] term.