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

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.