Question

An arithmetic progression starts with a positive fraction and every alternate term is an integer. If the sum of the first 11 terms is 33, then find the fourth term.

Answer 1

Hint: In this problem, first we need to apply the formula for sum of \[n\] terms in an A.P. Now, we need to consider first term equal to the common difference and hence find the fourth term of the A.P.

Complete step by step answer:
The formula for the sum \[S\] of \[n\] terms in A.P. is shown below.
\[S = \dfrac{n}{2}\left\{ {2a + \left( {n - 1} \right)d} \right\}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,......\left( 1 \right)\]
Here, \[a\] is the first term and \[d\] is a common difference.
Now, substitute 11 for \[n\] and 33 for \[S\] in equation (1).
\[
  \,\,\,\,\,33 = \dfrac{{11}}{2}\left\{ {2a + \left( {11 - 1} \right)d} \right\} \\
   \Rightarrow 66 = 11\left\{ {2a + 10d} \right\} \\
   \Rightarrow 2a + 10d = \dfrac{{66}}{{11}} \\
   \Rightarrow 2a + 10d = 6 \\
   \Rightarrow a + 5d = 3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,......\left( 2 \right) \\
\]
Since \[a\] is a fraction, \[d\] must be the same fraction in order to obtain sum \[a + d\] as an integer second term.
Now, substitute \[d\] for \[a\] in equation (2) to obtain the value of \[a\].
\[
  \,\,\,\,\,a + 5a = 3 \\
   \Rightarrow 6a = 3 \\
   \Rightarrow a = \dfrac{3}{6} \\
   \Rightarrow a = \dfrac{1}{2} \\
\]
The formula for the \[{n^{th}}\] term of an A.P. is shown below.
\[{T_n} = a + \left( {n - 1} \right)d\]
Substitute \[\dfrac{1}{2}\] for \[a\], \[\dfrac{1}{2}\] for \[d\] and for \[n\] in above formula.
\[
  {T_4} = \dfrac{1}{2} + \left( {4 - 1} \right)\dfrac{1}{2} \\
  {T_4} = \dfrac{1}{2} + \dfrac{3}{2} \\
  {T_4} = \dfrac{{1 + 3}}{2} \\
  {T_4} = 2 \\
\]

Thus, the fourth term of the A.P. is 2.

Note: Since, the first term is a fraction, common difference should be the same fraction, so that adding \[a + d\] gives a second integer term. The formula for the sum of \[n\] numbers in arithmetic progression is \[\dfrac{{n\left( {2a + \left( {n - 1} \right)d} \right)}}{2}\,\,\,{\text{or}}\,\,\,\dfrac{{n\left( {a + l} \right)}}{2}\], here a is first term, l is last term and d is common difference.