Question

In an arithmetic sequence, the third term is 27 and the eight term is 62. What is the 5th term in the sequence?

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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.

Given that,

$\red{\rm :\longmapsto\:a_3 = 27 \: }$

$\rm :\longmapsto\:a + (3 - 1)d = 27$

$\rm :\longmapsto\:a + 2d = 27 - - - (1)$

$\red{\rm :\longmapsto\:a_8 = 62 \: }$

$\rm :\longmapsto\:a + (8 - 1)d = 62$

$\rm :\longmapsto\:a + 7d = 62 - - - - (2)$

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

$\rm :\longmapsto\:5d = 35$

$\bf\implies \:d \: = \: 7$

On substituting d = 7 in equation (1), we get

$\rm :\longmapsto\:a + 2(7) = 27$

$\rm :\longmapsto\:a + 14 = 27$

$\rm :\longmapsto\:a = 27 - 14$

$\bf\implies \:a \: = \: 13$

Now,

$\rm :\longmapsto\:a_5$

$\rm \:  =  \:a + (5 - 1)d$

$\rm \:  =  \:a + 4d$

$\rm \:  =  \:13 + 4 \times 7$

$\rm \:  =  \:13 + 28$

$\rm \:  =  \:41$

$\rm \implies\:\boxed{ \tt{ \: a_5 \: = \: 41 \: }}$

More to explore :-

↝ 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.

Answer 2

Answer:

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

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.

Given that,

$\red{\rm :\longmapsto\:a_3 = 27 \: }$

$\rm :\longmapsto\:a + (3 - 1)d = 27$

$\rm :\longmapsto\:a + 2d = 27 - - - (1)$

$\red{\rm :\longmapsto\:a_8 = 62 \: }$

$\rm :\longmapsto\:a + (8 - 1)d = 62$

$\rm :\longmapsto\:a + 7d = 62 - - - - (2)$

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

$\rm :\longmapsto\:5d = 35$

$\bf\implies \:d \: = \: 7$

On substituting d = 7 in equation (1), we get

$\rm :\longmapsto\:a + 2(7) = 27$

$\rm :\longmapsto\:a + 14 = 27$

$\rm :\longmapsto\:a = 27 - 14$

$\bf\implies \:a \: = \: 13$

Now,

$\rm :\longmapsto\:a_5$

$\rm \:  =  \:a + (5 - 1)d$

$\rm \:  =  \:a + 4d$

$\rm \:  =  \:13 + 4 \times 7$

$\rm \:  =  \:13 + 28$

$\rm \:  =  \:41$

$\rm \implies\:\boxed{ \tt{ \: a_5 \: = \: 41 \: }}$

More to explore :-

↝ 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.