Question

If 6 times of term of an A. P equal to 8 times of its 8th term then show that on 14 th term​

Answer 1

$\begin{gathered}\begin{gathered}\bf Given -  \begin{cases} &\sf{6 \: a_6 \: = \: 8 \: a_8} \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf  To \:  show -   \begin{cases} &\sf{a_{14} \: = \: 0}  \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\Large{\bold{\pink{\underline{Formula \: Used \::}}}} \end{gathered}$

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.

$\large\underline\purple{\bold{Solution :- }}$

Let the first term of an AP series is 'a' and common difference is 'd'.

Tʜᴜs,

↝ 6ᵗʰ term is,

$\begin{gathered}\bf{a_6\:=\:a\:+\:(6\:-\:1)\:d} \\ \end{gathered}$

$\tt : \implies \: a_6 \: = \: a \: + \: 5d$

And

↝ 8ᵗʰ term is,

$\begin{gathered}\bf{a_8\:=\:a\:+\:(8\:-\:1)\:d} \\ \end{gathered}$

$\tt : \implies \: a_8 \: = \: a \: + 7d$

Now, According to statement,

$\tt : \implies \: 6 \: a_6 \: = \: 8 \: a_8$

$\tt : \implies \: 6 \: (a + 5d) = 8 \: (a + 7d)$

$\tt : \implies \: 6a \: + \: 30d = 8a \: + \: 56d$

$\tt : \implies \: 8a \: - 6a \: + 56d \: - \: 30d \: = \: 0$

$\tt : \implies \: 2a \: + \: 26d \: = \: 0$

$\tt : \implies \: a \: + \: 13d \: = \: 0$

$\tt : \implies \: a \: + \: (14 \: - \: 1)d \: = 0$

$\tt : \implies \: a_{14} \: = \: 0$

$\large{\boxed{\boxed{\bf{Hence, Proved}}}}$

Additional Information: -

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

Or

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

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.

Sₙ is the sum of  n term.