Find the sum of the first ten terms of the A.P. whose 4th term is -15 and 9th term is -30

Answer

The sum of the first 10 terms if the given A.P. is -195.

Explanation

Let the first term of the A.P. be a and the common difference d. Here, `t_(4)=-15, t_(9)=-30`
`t_(n)=a+(n-1)d` …(Formula)
`thereforet_(4)=a+(n-1)d`
`therefore-15=a+3d" "thereforea+3d=-15 ...(1)`
Also,`t_(9)=a+(9-1)d`
`therefore-30=a+8d" "thereforea+8d=-30" "...(2)`
Subtracting equation (1) from equation (2),
`a+8d=-30` ...(2)
`a+3d=-15` ...(1)
`(---+)/(" "5d=-15)" "therefored=-3`
Subtracting `d=-3` in equation (1),
`a+3(-3)=-15" "thereforea-9=-15`
`thereforea=-15+9 " "thereforea=-6`
Let us find the sum of the first 10 terms.
`S_(n)=(n)/(2)[2a+(n-1)d]` ...(Formula)
`thereforeS_(10)=(10)/(2)[2axx(-6)+(10-1)xx(-3)]` ...(Substituting the values)
`=5[-12+9xx(-3)]`
`=5(-12-27)`
`=5xx(-39)`
`thereforeS_(10=-195`