Suppose that \(\{a_n\}\)is an arithmetic sequence with
\(a_1+a_2+ \cdots +a_{100}=100 \quad \text{and} \quad a_{101}+a_{102}+ \cdots + a_{200}=200.\)
what is the value of \(a_2 - a_1\)? Express your answer as a common fraction.
Sum =N/2*[2F + (N - 1) * D, where N=Number of terms, F=First term, D=Common difference.
First term =101/200, Common difference =1/100, Number of terms =100
Sum =100/2*[(2*101/200) + (100 - 1) * 1/100]
Sum =50 * [(202/200) + (99) * 1/100]
Sum = 50 * [1.01 + 0.99]
Sum =50 * 2
Sum=100 - sum of the 1st. sequence
Second sequence = 301/200 =F, Common difference =1/100, Number of terms =100
Sum = 100/2 * [(2*301/200) + (99) * 1/100]
Sum = 50 * [(602/200) + 0.99]
Sum = 50 * [3.01 + 0.99]
Sum = 50 * 4
Sum = 200 - sum of the 2nd. sequence.
So, a(2) - a(1) =(103/200 - 101/200) =1/100
