Question

Find the middle term of the AP 6, 13, 20, ……, 216.

Answer 1

Hint: Compare the given AP with the general AP to get the values of first term and common difference, now use the nth term formula to find the number of terms , depending on the number of terms whether odd or even use the middle term formula.

Complete step-by-step answer:

The given series of an arithmetic progression (AP) is 6, 13, 20, ……, 216.
We know that the general series of arithmetic progression is given by $a,\text{ }a+d,\text{ }a+2d,..a+\left( n-1 \right)d$
Where a, d and n are the first term, the common difference of the series of arithmetic progression and the number of terms in the series of an arithmetic progression.
By comparing the terms of both the series we get,
The first term of arithmetic progression = a = 6
The second term of arithmetic progression = a + d = 13
The last term of arithmetic progression = a+(n-1)d = 216
By substituting the first term value in the second term of arithmetic progression we get,
6 + d = 13
$\Rightarrow$d = 13 – 6 = 7
By substituting the values of a, d in the last term of arithmetic progression we get,
$\Rightarrow$ 6 + 7(n-1) = 216
$\Rightarrow$ 6 + 7n – 7 = 216
$\Rightarrow$ 7n - 1 = 216
$\Rightarrow$ 7n = 216 + 1
$\Rightarrow$ 7n = 217
$\Rightarrow n=\dfrac{217}{7}=31$
Thus, the number of terms of the given arithmetic progression = 31
We know that the middle term of a progression is ${{\left( \dfrac{n+1}{2} \right)}^{th}}$term of the series when n is odd and equal to ${{\left( \dfrac{n}{2} \right)}^{th}}$term of the series when n is even.
The middle term of the given arithmetic progression is ${{\left( \dfrac{31+1}{2} \right)}^{th}}$ term since 31 is an odd number.
$={{\left( \dfrac{32}{2} \right)}^{th}}={{16}^{th}}$term of the series
We know that ${{n}^{th}}$term of the series is given by the formula a + (n - 1)d
$\Rightarrow {{16}^{th}}$term is equal to a + (16 - 1)d = a + 15d
By substituting the values of a, d in the ${{16}^{th}}$ term of arithmetic progression we get,
$=6+15\times 7$
$= 6 + 105$
$\Rightarrow {{16}^{th}}$term = 111
$\therefore$ The middle term of the given arithmetic progression = 111

Note: The possibility of mistake can be not using the arithmetic progression general series formula to compare and calculate the answer. Using another method of writing series using a pattern method will elongate the process of solving.