Question

Ths seventh term of an arithmetic sequence is 21. a)What is the sum of sixth and eighth term of this sequence? b) what is the sum of first and thirteenth term?​

Answer 1

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

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ nᵗʰ term of an arithmetic sequence is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{a_n\:=\:a\:+\:(n\:-\:1)\:d}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

• aₙ is the nᵗʰ term.

• a is the first term of the sequence.

• n is the no. of terms.

• d is the common difference.

Tʜᴜs,

According to statement,

$\rm :\longmapsto\:a_7 = 21$

$\rm :\longmapsto\:a + (7 - 1)d = 21$

$\bf :\longmapsto\:a + 6d = 21 - - - - (1)$

Now, Consider

$\green{\bf :\longmapsto\:a_6 + a_8}$

$\rm \: = \: \:a + (6 - 1)d + a + (8 - 1)d$

$\rm \: = \: \:a + 5d + a + 7d$

$\rm \: = \: \:2a + 12d$

$\rm \: = \: \:2(a + 6d)$

$\rm \: = \: \:2 \times 21$

$\rm \: = \: \:42$

Hence,

$\green{\boxed{ \bf{ \: \bf :\longmapsto\:a_6 + a_8 = 42 \qquad}}}$

Now, Consider

$\blue{\bf :\longmapsto\: a_1+ a_{13}}$

$\rm \: = \: \:a + a + (13 - 1)d$

$\rm \: = \: \:2a + 12d$

$\rm \: = \: \:2(a + 6d)$

$\rm \: = \: \:2 \times 21$

$\rm \: = \: \:42$

Hence,

$\blue{\boxed{ \bf{ \: \bf :\longmapsto\:a_1 + a_{13} = 42 \qquad}}}$

Additional Information :-

↝ Sum of n  terms of an arithmetic sequence is

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{S_n\:=\dfrac{n}{2} \bigg(2 \:a\:+\:(n\:-\:1)\:d \bigg)}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

• Sₙ is the sum of n terms of AP.

• a is the first term of the sequence.

• n is the no. of terms.

• d is the common difference.