Question

Find the middle term of the AP \[10,7,4,.........( - 62)\].

Answer 1

Hint: Whenever the number of terms is not given in the question we will calculate the number of terms in AP i.e. arithmetic progression first. Then on the basis of number of terms i.e. even number of terms or odd number of terms,
We calculate the middle term of the given AP series.

Formula used: To calculate number of terms, we use that his formula: \[l{\text{ }} = {\text{ }}a + \left( {{\text{ }}n - 1{\text{ }}} \right)d\]
Where n is our number of terms, l is our last term, a is our first term and d is the common difference between two consecutive terms.
If n is even, Middle terms are\[{\left( {\dfrac{n}{2}} \right)^{th}}\]and \[{\left( {\dfrac{n}{2} + 1} \right)^{th}}\] term.
If n is odd, Middle terms will be\[{\left( {\dfrac{{n + 1}}{2}} \right)^{th}}\]term.
Nth term is =\[a + \left( {{\text{ }}n - 1{\text{ }}} \right)d\]

Complete step by step solution: First we will calculate number of terms from the above given formula:
\[a = 10\][Given in the question]
\[d = - 3\]
\[l = - 62\]
Hence \[ - 62 = 10 + (n - 1)( - 3)\]
\[ \Rightarrow - 62 - 10 = (n - 1)( - 3)\]
Or \[\dfrac{{ - 72}}{{ - 3}} = (n - 1)\]
Or \[(n - 1) = 24\]
Or \[n = 24 + 1 = 25\]
Clearly n is odd here, hence our middle term will be \[{\left( {\dfrac{{n + 1}}{2}} \right)^{th}}\]term
i.e.\[\dfrac{{25 + 1}}{2} = \dfrac{{26}}{2} = {13^{th}}\] term will be our middle term.
Now calculate\[{13^{th}}\]term, this is calculated as
${a_{13}} = 10 + (13 - 1)( - 3)$
$ \Rightarrow {a_{13}} = 10 + 12( - 3)$
${a_{13}} = 10 - 36 = - 26$

Our desired answer is -26.

Note: We can easily calculate the middle term if our AP series is finite but when AP series is not finite most students get confused about where they will start calculating. So in an Arithmetic Progression series if a number of terms are not given, calculate it first. Also take care while calculating the common difference of arithmetic progression which is indicated by d and given as \[d = {a_n} - {a_{(n - 1)}}\]. Most students directly subtract first term and second term, due to which sign error will occur. So be careful while calculating d. Students must be clear about d, that common difference may be positive or negative.