Answer
${a_1} = 1 + \left( {2\cdot1} \right) = 3$
${a_2} = 2 + \left( {2 + 1} \right) + \left( {2\cdot2} \right) = 9$
${a_3} = 3 + \left( {3 + 1} \right) + \left( {3 + 2} \right) + \left( {2\cdot3} \right) = 18$
${a_4} = 4 + \left( {4 + 1} \right) + \left( {4 + 2} \right) + \left( {4 + 3} \right) + \left( {2\cdot4} \right) = 30$
Work Step by Step
The sequence:
${a_n} = n + \left( {n + 1} \right) + \left( {n + 2} \right) + ... + \left( {2n} \right)$ for $n=1,2,3,4$
For $n=1$, the last term is 2. So,
${a_1} = 1 + \left( {2\cdot1} \right) = 3$
For $n=2$, the last term is 4. So,
${a_2} = 2 + \left( {2 + 1} \right) + \left( {2\cdot2} \right) = 9$
For $n=3$, the last term is 6. So,
${a_3} = 3 + \left( {3 + 1} \right) + \left( {3 + 2} \right) + \left( {2\cdot3} \right) = 18$
For $n=4$, the last term is 8. So,
${a_4} = 4 + \left( {4 + 1} \right) + \left( {4 + 2} \right) + \left( {4 + 3} \right) + \left( {2\cdot4} \right) = 30$
