Question

Find the sums of: -5+(-8)+(-11)+...+(-230).
Chapter AP and GP.​

Answer 1

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

Given series is

$\rm :\longmapsto\: - 5 + ( - 8) + ( - 11) + - - - - + ( - 230)$

Its an AP series, with

• First term, a = - 5

• Common difference, d = (-8) - (-5) = - 3

• nᵗʰ term of series, aₙ = - 230

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,

On substituting the values in above formula, we get

$\rm :\longmapsto\: - 230 = - 5 + (n - 1)( - 3)$

$\rm :\longmapsto\: - 230 + 5 = - 3n + 3$

$\rm :\longmapsto\: - 225 = - 3n + 3$

$\rm :\longmapsto\: - 225 - 3 = - 3n$

$\rm :\longmapsto\: - 228 = - 3n$

$\bf\implies \:n = 76$

Further,

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ Sum of first n terms of an arithmetic sequence is,

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

Wʜᴇʀᴇ,

• aₙ is the nᵗʰ term.

• a is the first term of the sequence.

• n is the no. of terms.

Tʜᴜs,

On substituting the values in above formula, we get

$\rm :\longmapsto\:S_n = \dfrac{76}{2} ( - 5 - 230)$

$\rm :\longmapsto\:S_n = 38 \times ( - 235)$

$\bf\implies \:S_n = - \: 8930$

Additional Information :-

↝ Sum of first n terms of an arithmetic sequence is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{S_n\:=\dfrac{n}{2} \bigg(\:2a\:+\:(n\:-\:1)\:d \bigg)}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

• a is the first term of the sequence.

• n is the no. of terms.

• d is the common difference.