Question

Use the values of a_1 and S_n to find the value of a_n.

Answer 1

Given :

`\begin{gathered} a_1=_{}10\text{ } \\ S_(30)\text{ = }4350 \end{gathered}`

Calculation :

The sum of the arithmetic progression is given as,

`\begin{gathered} S_n=\text{ }(n)/(2)\lbrack2a_1\text{ + ( n - 1 )d \rbrack} \\ S_(30)\text{ =}(30)/(2)\text{ \lbrack 2}*\text{ 10 + ( 30 - 1 ) d \rbrack} \\ \end{gathered}`

Further,

`\begin{gathered} 4350\text{ = 15 }*\text{ \lbrack 20 + 29d \rbrack} \\ 20\text{ + 29 d = }(4350)/(15) \\ 20\text{ + 29d = }290 \\ 29d\text{ = 290 - 20} \end{gathered}`

Therefore,

`\begin{gathered} d\text{ = }(270)/(29) \\ d\text{ = 9.31} \end{gathered}`

The 30th term of the given sequence is calculated as,

`\begin{gathered} a_(30)=a_1\text{ + ( 30 - 1 )9.31} \\ a_{30\text{ }}=\text{ 10 + 29 }*\text{ 9.31} \\ a_{30\text{ }}=\text{ 10 + 269.99} \\ a_{30\text{ }}=\text{ 280} \end{gathered}`

Thus the 30th term of the given sequence is 280.