Question

For the arthmetic series (A.S.) 2+6+10+14+.......+98, find the number of terms n and the sum of the series S(n)

Answer 1

Number of terms:

1. Find the common difference: Subtract the first term from the second:

`6-2=4`

2. Us the last term (tn), first term (a1), and common difference (d) in the next equation to solve (number of terms (n):

`t_n=a_1+(n-1)d`

`\begin{gathered} 98=2+(n-1)4 \\ \text{Subtract 2 in both sides of the equation:} \\ 98-2=2-2+(n-1)4 \\ 96=(n-1)4 \\ \\ \text{Divide both sides of the equation into 4:} \\ (96)/(4)=((n-1)4)/(4) \\ \\ 24=n-1 \\ \\ Add\text{ 1 in both sides of the equation:} \\ 24+1=n-1+1 \\ 25=n \\ \\ \\ n=25 \end{gathered}`Number of terms: 25

Sum of the terms:

`S_n=(n(a_1+a_n))/(2)`
`\begin{gathered} S_{\mleft\{25\mright\}}=(25(2+98))/(2) \\ \\ S_{\mleft\{25\mright\}}=(25(100))/(2) \\ \\ S_{\{25\}}=(2500)/(2) \\ \\ S_{\{25\}}=1250 \end{gathered}`Sum of the series: 1250