Question

the middle term of AP 6,13,20 ..... 216 is​

Answer 1

$\sf\large\underline{Given:}$

• $\rm{The\: first\: term (a)=6}$
• $\rm{The\:last\: term (Tn)=216}$
• $\rm{The\: common\: difference=7}$

$\sf\large\underline{To\: Find:}$

• $\rm{The\: middle\: term\:of\:AP=?}$

$\sf\large\underline{Solution:}$

• [At 1st calculate the number n term than calculate median (means the middle term of AP)]

$\sf\large\underline{Formula\: used:}$

$\rm\purple{\implies T_{n}=a+(n-1)d}$

$\sf\large\underline{A.T.Q:}$

$\rm{\implies 216=6+(n-1)7}$

$\rm{\implies 216-6=(n-1)7}$

$\rm{\implies 210=(n-1)7}$

$\rm{\implies 30=n-1}$

$\rm{\implies n=30+1}$

$\rm\red{\implies n=31}$

• [Now, calculate middle term here]
• Total number of n term=31

$\rm{\implies M=(\dfrac{n+1}{2})th}$

$\rm{\implies M=(\dfrac{31+1}{2})th}$

$\rm{\implies M=(\dfrac{32}{2})th}$

$\rm{\implies M=16th}$

• {Than find out T_16 here}

$\rm{\implies T_{16}=6+(16-1)7}$

$\rm{\implies T_{16}=6+105}$

$\rm{\implies T_{16}=111}$

Hence,

$\rm\red{\implies The\: middle\:term=111}$

Answer 2

Question :–

▪︎ Find the middle term of A.P. 6 , 13 , 20 , ..... , 216.

ANSWER :–

GIVEN :–

$\\ { \bold{ \: \: . \: \: First \: \: term \: \: of \: \:A.P. = 6}}$

$\\ { \bold{ \: \: . \: \: Common \: \: difference \: \: of \: \:A.P. = 7}}$

$\\ { \bold{ \: \: . \: \: Last \: \: term \: \: of \: \: A.P. = 216}}$

TO FIND :–

• Middle term of given series.

SOLUTION :–

▪︎ We know that nth term of A.P. –

$\\ \implies \large { \red{ \boxed{ \bold{ T_{n} = a + (n - 1)d }}}}$

▪︎ Now put the values –

$\\ \implies { \bold{ 216 = 6 + (n - 1)(7) }}$

$\\ \implies { \bold{ 216 = 6 +7n - 7 }}$

$\\ \implies { \bold{ 216 =7n - 1 }}$

$\\ \implies { \bold{ n = \cancel \frac{217}{7} }}$

$\\ \implies { \boxed { \bold{ n = 31 }}}$

$\\ \longrightarrow { \bold{ Middle \: \: term = \frac{n + 1}{2} \th term }}$

$\\ \longrightarrow { \bold{ Middle \: \: term = \frac{31 + 1}{2} }}$

$\\ \longrightarrow { \bold{ Middle \: \: term = \cancel \frac{32}{2} }}$

$\\ \longrightarrow { \bold{ Middle \: \: term = 16 }}$

▪︎ So that , Middle term is 16th term.

$\\ \implies { \bold{ 16 \th \: \: term( T_{16}) = a + (16 - 1)d }}$

$\\ \implies { \bold{ T_{16} = 6 + (15)(7) }}$

$\\ \implies { { \bold{ T_{16} = 6 + 105 }}}$

$\\ \implies \large { \red { \boxed { \bold{ T_{16} = 111 }}}}$

Hence , The Middle term is 111