if the sum of fist 6 terms of ap is 96 and sum of first 10 term is 240 ,then find the sum of 20 term of ap​

1 thought on “if the sum of fist 6 terms of ap is 96 and sum of first 10 term is 240 ,then find the sum of 20 term of ap​”

1. [tex]\large\underline{\sf{Solution-}}[/tex]

   Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

   ↝ Sum of n terms of an arithmetic sequence is,

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

   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.

   Tʜᴜs,

   According to statement,

   Given that

   ☆ Sum of first 6 terms of an AP is 96.

   [tex]\rm :\longmapsto\:S_6 = 96[/tex]

   [tex]\rm :\longmapsto\:\dfrac{6}{2} \bigg(2 \:a\:+\:(6\:-\:1)\:d \bigg) = 96[/tex]

   [tex]\rm :\longmapsto\:2a + 5d = 32 – – – (1)[/tex]

   Also,

   Given that

   ☆ Sum of 10 terms is 240

   [tex]\rm :\longmapsto\:S_{10} = 240[/tex]

   [tex]\rm :\longmapsto\:\dfrac{10}{2} \bigg(2 \:a\:+\:(10\:-\:1)\:d \bigg) = 240[/tex]

   [tex]\rm :\longmapsto\:2a + 9d = 48 – – – (2)[/tex]

   ○ On Subtracting equation (1) from equation (2), we get

   [tex]\rm :\longmapsto\:4d = 16[/tex]

   [tex]\bf\implies \:d = 4[/tex]

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

   [tex]\rm :\longmapsto\:2a + 5 \times 4 = 32[/tex]

   [tex]\rm :\longmapsto\:2a + 20 = 32[/tex]

   [tex]\rm :\longmapsto\:2a = 32 – 20[/tex]

   [tex]\rm :\longmapsto\:2a = 12[/tex]

   [tex]\bf\implies \:a = 6[/tex]

   So,

   Now we have ,

   • First term of AP, a = 6

   • Common difference of AP, d = 4

   • Number of terms, n = 20

   Sum of 20 terms, is

   [tex]\rm :\longmapsto\:S_{20}\:=\dfrac{20}{2} \bigg(2 \times 6\:+\:(20\:-\:1)\:4 \bigg)[/tex]

   [tex] \rm \: \: = \: \: 10(12 + 76)[/tex]

   [tex] \rm \: \: = \: \: 10(88)[/tex]

   [tex] \rm \: \: = \: \: 880[/tex]

   [tex]\bf\implies \:S_{20} = 880[/tex]

   Additional Information :-

   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.

   Reply