Question

The 8 term of an AP is 31 and its 15 term is 16 more than the 11 term.
Find the first 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,

↝ 8ᵗʰ term of an AP is 31

$\rm :\longmapsto\:a + (8 - 1)d = 31$

$\rm :\longmapsto\:a + 7d = 31 - - - (1)$

According to statement again,

↝ 15ᵗʰ term of an AP is 16 more than its 11ᵗʰ term.

$\rm :\longmapsto\:a + (15 - 1)d = 16 + a + (11 - 1)d$

$\rm :\longmapsto\:a + 14d = 16 + a + 10d$

$\rm :\longmapsto\:4d = 16$

$\bf :\implies\:d = 4 \: - - (2)$

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

$\rm :\longmapsto\:a + 7 \times 4 = 30$

$\rm :\longmapsto\:a + 28 = 31$

$\bf\implies \:a = 3$

Additional Information :-

1. Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝Sum of first 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 first n terms

• a is the first term of the sequence.

• n is the no. of terms.

• d is the common difference.

2. Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ nᵗʰ term from the end of an arithmetic sequence is,

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

Wʜᴇʀᴇ,

• aₙ is the nᵗʰ term.

• l is the last term of the sequence.

• n is the no. of term.

• d is the common difference.