[tex]\large\underline{\sf{Solution-}}[/tex] Given series is [tex]\rm :\longmapsto\: – 5 + ( – 8) + ( – 11) + – – – – + ( – 230)[/tex] 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, [tex]\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{a_n\:=\:a\:+\:(n\:-\:1)\:d}}}}}} \\ \end{gathered}[/tex] 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 [tex]\rm :\longmapsto\: – 230 = – 5 + (n – 1)( – 3)[/tex] [tex]\rm :\longmapsto\: – 230 + 5 = – 3n + 3[/tex] [tex]\rm :\longmapsto\: – 225 = – 3n + 3[/tex] [tex]\rm :\longmapsto\: – 225 – 3 = – 3n[/tex] [tex]\rm :\longmapsto\: – 228 = – 3n[/tex] [tex]\bf\implies \:n = 76[/tex] Further, Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ, ↝ Sum of first n terms of an arithmetic sequence is, [tex]\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{S_n\:=\dfrac{n}{2} (\:a\:+ \:a_n )}}}}}} \\ \end{gathered}[/tex] 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 [tex]\rm :\longmapsto\:S_n = \dfrac{76}{2} ( – 5 – 230)[/tex] [tex]\rm :\longmapsto\:S_n = 38 \times ( – 235)[/tex] [tex]\bf\implies \:S_n = – \: 8930[/tex] Additional Information :- ↝ Sum of first n terms of an arithmetic sequence is, [tex]\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{S_n\:=\dfrac{n}{2} \bigg(\:2a\:+\:(n\:-\:1)\:d \bigg)}}}}}} \\ \end{gathered}[/tex] Wʜᴇʀᴇ, a is the first term of the sequence. n is the no. of terms. d is the common difference. Reply
[tex]\large\underline{\sf{Solution-}}[/tex]
Given series is
[tex]\rm :\longmapsto\: – 5 + ( – 8) + ( – 11) + – – – – + ( – 230)[/tex]
Its an AP series, with
Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,
↝ nᵗʰ term of an arithmetic sequence is,
[tex]\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{a_n\:=\:a\:+\:(n\:-\:1)\:d}}}}}} \\ \end{gathered}[/tex]
Wʜᴇʀᴇ,
Tʜᴜs,
On substituting the values in above formula, we get
[tex]\rm :\longmapsto\: – 230 = – 5 + (n – 1)( – 3)[/tex]
[tex]\rm :\longmapsto\: – 230 + 5 = – 3n + 3[/tex]
[tex]\rm :\longmapsto\: – 225 = – 3n + 3[/tex]
[tex]\rm :\longmapsto\: – 225 – 3 = – 3n[/tex]
[tex]\rm :\longmapsto\: – 228 = – 3n[/tex]
[tex]\bf\implies \:n = 76[/tex]
Further,
Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,
↝ Sum of first n terms of an arithmetic sequence is,
[tex]\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{S_n\:=\dfrac{n}{2} (\:a\:+ \:a_n )}}}}}} \\ \end{gathered}[/tex]
Wʜᴇʀᴇ,
Tʜᴜs,
On substituting the values in above formula, we get
[tex]\rm :\longmapsto\:S_n = \dfrac{76}{2} ( – 5 – 230)[/tex]
[tex]\rm :\longmapsto\:S_n = 38 \times ( – 235)[/tex]
[tex]\bf\implies \:S_n = – \: 8930[/tex]
Additional Information :-
↝ Sum of first n terms of an arithmetic sequence is,
[tex]\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{S_n\:=\dfrac{n}{2} \bigg(\:2a\:+\:(n\:-\:1)\:d \bigg)}}}}}} \\ \end{gathered}[/tex]
Wʜᴇʀᴇ,