Question

Find the sum of all 2-digit numbers which are divisible by 3.

Answer 1

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

Let first find the series of the two digit number which are divisible by 3.

$\rm :\longmapsto\:12, \: 15, \: 18, - - - - ,99$

Now, it forms an AP series, with

$\rm :\longmapsto\:first \: term, \: a = 12$

$\rm :\longmapsto\:common \: difference, \: d = 3$

$\rm :\longmapsto\: {n}^{th} \: term, \: a_n = 99$

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,

$\rm :\longmapsto\:99 = 12 + (n - 1)3$

$\rm :\longmapsto\:99 - 12 = (n - 1)3$

$\rm :\longmapsto\:87 = (n - 1)3$

$\rm :\longmapsto\:29 = n - 1$

$\bf\implies \:\boxed{ \bf{ \:n \: = \: 30}}$

Now,

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ Sum of n  terms of an arithmetic sequence is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{S_n\:=\dfrac{n}{2} \bigg( \:a\:+ \: a_n \bigg)}}}}}} \\ \end{gathered}$

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,

$\rm :\longmapsto\:S_n \: = \: \dfrac{30}{2}(12 + 99)$

$\rm :\longmapsto\:S_n \: = \: 15 \times 111$

$\rm :\longmapsto\:\boxed{ \bf{ \:S_n \: = \: 1665}}$

Additional Information :-

↝ 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.