Question

Sum of 25 terms of an arithmetic sequence is225 what is it 13th term

Answer 1

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

Given that,

Sum of 25 terms of an arithmetic sequence is 225

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, on substituting the values, we get

$\rm :\longmapsto\:225 = \dfrac{15}{2} \bigg(2a + (25 - 1)d\bigg)$

$\rm :\longmapsto\:15 = \dfrac{1}{2} \bigg(2a + 24d\bigg)$

$\rm :\longmapsto\:15 = \dfrac{2}{2} \bigg(a + 12d\bigg)$

$\rm \implies\:\boxed{\tt{ a + 12d = 15 \: }} - - - (1)$

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,

$\red{\rm :\longmapsto\:a_n\:=\:a\:+\:(n\:-\:1)\:d}$

$\red{\rm :\longmapsto\:a_{13}\:=\:a\:+\:(13\:-\:1)\:d}$

$\red{\rm :\longmapsto\:a_{13}\:=\:a\:+\:12\:d}$

$\red{\rm :\longmapsto\:a_{13}\:=\:15\:}$