The sum of first 15th term of an arithmetic sequence is 570 and its 12

1.	The sum of first 15th term of an arithmetic sequence is 570 and its 12th trem is 62What is the sum of 30th term
Answer» $\large\underline{\sf{Solution-}}$ We know that ↝ 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. Now, Given that Sum of first 15 terms of an AP = 570 $\red{\rm :\longmapsto\:S_{15}\:=\dfrac{15}{2} \bigg(2 \:a\:+\:(15\:-\:1)\:d \bigg)}$ $\rm :\longmapsto\:570 = \dfrac{15}{2}(2a + 14d)$ $\rm :\longmapsto\:38 = \dfrac{2}{2}(a + 7d)$ $\bf\implies \:a + 7d = 38 - - - (1)$ Also, we know that ↝ 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. Now, Given that $\rm :\longmapsto\:a_{12}\:=62$ $\rm :\longmapsto\:a + (12 - 1)d = 62$ $\rm :\longmapsto\:a + 11d = 62 - - - (2)$ On SUBTRACTING equation (1) from equation (2), we get $\rm :\longmapsto\:4d = 24$ $\bf\implies \:d = 6$ On SUBSTITUTING the value of d, in equation (1) we get $\rm :\longmapsto\:a + 7 \times 6 = 38$ $\rm :\longmapsto\:a + 42= 38$ $\rm :\longmapsto\:a = 38 - 42$ $\bf\implies \:a = - 4$ Now, Sum of first 30 terms is given by $\rm :\longmapsto\:S_{30}\:=\dfrac{30}{2} \bigg(2 \:( - 4)\:+\:(30\:-\:1)\:6 \bigg)$ $\rm :\longmapsto\:S_{30}\:=15 \bigg( - \:8\:+\:(29)\:6 \bigg)$ $\rm :\longmapsto\:S_{30}\:=15 \bigg( - \:8\:+\:174 \bigg)$ $\rm :\longmapsto\:S_{30}\:=15 \times 166$ $\bf :\longmapsto\:S_{30}\:=2490$

The sum of first 15th term of an arithmetic sequence is 570 and its 12th trem is 62What is the sum of 30th term

Answer»

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

We know that

↝ Sum of n TERMS of an arithmetic sequence is,

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