Question

please help me.
please don't scam.​

Answer 1

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

Given that,

First term of an AP, a = 4

Also, given that

Sum of the first 5 terms is equals to one - fourth of the sum of next 5 terms.

It implies,

$\rm :\longmapsto\:S_5 = \dfrac{1}{4}[S_{10} - S_5]$

$\rm :\longmapsto\:4S_5 = S_{10} - S_5$

$\rm :\longmapsto\:5S_5 = S_{10}$

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ 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.

So,

$\rm :\longmapsto\:5 \times \dfrac{5}{2} \bigg[2(4) + (5 - 1)d\bigg] = \dfrac{10}{2}\bigg[2(4) + (10 - 1)d\bigg]$

$\rm :\longmapsto\: \dfrac{5}{2} \bigg[8 + 4d\bigg] = \bigg[8 + 9d\bigg]$

$\rm :\longmapsto\: 5 \bigg[8 + 4d\bigg] = 2\bigg[8 + 9d\bigg]$

$\rm :\longmapsto\: 40 + 20d=16 + 18d$

$\rm :\longmapsto\: 20d - 18d = 16 - 40$

$\rm :\longmapsto\: 2d = - 24$

$\rm \implies\:\boxed{ \tt{ \: d \: = \: - \: 12 \: }}$

Now,

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,

$\rm :\longmapsto\:a_{12} = a + (12 - 1)d$

$\rm :\longmapsto\:a_{12} = a + 11d$

On substituting the values of a and d, we get

$\rm :\longmapsto\:a_{12} = 4 + 11( - 12)$

$\rm :\longmapsto\:a_{12} = 4 - 132$

$\rm :\longmapsto\:\boxed{ \tt{ \: a_{12} = - 128 \: }}$