Sample Question 1:

What is the value in simplest form of the following expression?$\sqrt{1} + \sqrt{1+3} + \sqrt{1+3+5} + \sqrt{1+3+5+7}$

$\textbf{(A) }5 \qquad \textbf{(B) }4 + \sqrt{7} + \sqrt{10} \qquad \textbf{(C) } 10 \qquad \textbf{(D) } 15 \qquad \textbf{(E) } 4 + 3\sqrt{3} + 2\sqrt{5} + \sqrt{7}$

---

Answer Keys

Question 1: C

---

Solutions

Question 1
Step 1: Take note of the given expression $\sqrt{1} + \sqrt{1+3} + \sqrt{1+3+5} + \sqrt{1+3+5+7}$.

Step 2: Simplify each square root in the expression by adding up the numbers under each square root: this gives $\sqrt{1} + \sqrt{4} + \sqrt{9} + \sqrt{16}$.

Step 3: Now, calculate the square root of each number: this results in $1 + 2 + 3 + 4$.

Step 4: Add those numbers together for a final result of $10$.

Step 5: So, the value in simplest form of the given expression is $\boxed{\textbf{(C)} 10}$.

Note: The pattern observed here is that the sum of the first 'n' odd numbers is $n^2$. This can be helpful in quickly simplifying expressions in a similar format.