Question

There are \[15\] terms in an arithmetic progression. Its first term is \[5\] and its sum is\[390\]. The middle term is
a) 23
b) 26
c) 29
d) 32

Answer 1

Hint:Using the general formula of the sum of n terms of AP which is given as\[{S_n} = \dfrac{n}{2}[2a + (n - 1)d]\] and by substituting the given values, we will solve for common difference d and then we will find the value of the middle term i.e. the eighth term using the formula\[{t_n} = a + (n - 1)d\]. Remember the value of n will be different in both formulas.

Formula Used:
\[{S_n} = \dfrac{n}{2}[2a + (n - 1)d]\]
\[{t_n} = a + (n - 1)d\].

Complete step by step Solution:
Given values: Number of terms (n) =\[15\]
First-term of arithmetic progression (a) =\[5\]
Sum of 15 terms \[{S_{15}} = 390\]
Use the formula of sum of n terms in Arithmetic Progression i.e.
\[{S_n} = \dfrac{n}{2}[2a + (n - 1)d]\]
Here\[{S_n} = 390\] ,\[n = 15\],\[a = 5\]
Substitute the given values in the above formula and get the value of common difference d
\[390 = \dfrac{{15}}{2}[2 \times 5 + (15 - 1)d]\]
Simplify it by opening the brackets
\[390 = \dfrac{{15}}{2}[10 + 14d]\]
\[390 \times 2 = 15\,[10 + 14d]\]
\[780 = 15\,[10 + 14d]\]
\[\dfrac{{780}}{{15}} = \,[10 + 14d]\]

\[52=10+14d\]
Solve for d
\[52 - 10 = 14d\]
\[42 = 14d\]
Therefore, \[d = \dfrac{{42}}{{14}} = 3\]
And now we will find the value of the middle term. In this question, the number of terms i.e. \[n = 15\] is an odd number. The formula to find the middle term in case n is an odd number is given by \[\dfrac{n+1}{2}\] . Put \[n = 15\] to get the middle term
 \[\dfrac{{15 + 1}}{2} = \dfrac{{16}}{2} = {8^{th}}\]
The middle of the given sequence is \[{8^{th}}\] term. To find the value of \[{8^{th}}\] term of the given series, substitute given values in the formula\[{t_n} = a + (n - 1)d\]. Remember the value of n will be 8 in this formula.
Put \[a=5\]],\[n = 8\] and \[d = 3\] in the formula \[{t_n} = a + (n - 1)d\]
\[{t_8} = 5 + (8 - 1) \times 3\]
\[{t_8} = 5 + 21\]
Therefore, \[t_s=26\]
The middle term is\[26\].

Hence, the correct option is b.

Note: While solving for the middle term of AP using the formula\[{t_n} = a + (n - 1)d\], the value of n should be taken as 8 since the eighth term is the middle term of the given arithmetic progression.