Question

The sum of first 7term of an arithmetic sequence 119 and sum of first 20terms is 860 and find its fourth term

Answer 1

Answer:

Step-by-step explanation:

Answer 2

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

Let assume that

First term of an AP series = a

Common difference of an AP series = d

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.

Thus, According to statement

$\rm :\longmapsto\:S_7 = 119$

$\rm :\longmapsto\:\dfrac{7}{2} \bigg(2 \:a\:+\:(7\:-\:1)\:d \bigg) = 119$

$\rm :\longmapsto\:\dfrac{1}{2} \bigg(2 \:a\:+\:(6)\:d \bigg) =17$

$\rm \implies\:\boxed{ \tt{ \: 2a + 6d = 34 \: }} - - - - (1)$

Also, Given that,

$\rm :\longmapsto\:S_{20} = 860$

$\rm :\longmapsto\:\dfrac{20}{2} \bigg(2 \:a\:+\:(20\:-\:1)\:d \bigg) = 860$

$\rm :\longmapsto\:2 \:a\:+ \: 19\:d= 86$

$\rm \implies\:\boxed{ \tt{ \: 2a + 19d = 86 \: }} - - - - (2)$

On Subtracting equation (1) from equation (2), we get

$\rm :\longmapsto\:13d = 52$

$\rm \implies\:\boxed{ \tt{ \: d \: = \: 4 \: }}$

On Substituting d = 4, in equation (1), we get

$\rm :\longmapsto\:2a + 6 \times 4 = 34$

$\rm :\longmapsto\:2a + 24= 34$

$\rm :\longmapsto\:2a = 34 - 24$

$\rm :\longmapsto\:2a = 10$

$\rm \implies\:\boxed{ \tt{ \: a \: = \: 5 \: }}$

Now,

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

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,

$\rm :\longmapsto\:a_4 = a + (4 - 1)d$

$\rm :\longmapsto\:a_4 = a + 3d$

$\rm :\longmapsto\:a_4 = 5 + 3(4)$

$\rm :\longmapsto\:a_4 = 5 + 12$

$\red{\rm \implies\:\boxed{ \tt{ \: a_4 \: = \: 17 \: }}}$