Question

Find $P_1 + P_2 + \dots + P_{10}$ if $P_k$ is the perimeter of a square having side length $k$. 
 

Hint:

Whenever perimeter depends linearly on $k$, factor out constants first and then use the formula for sum of natural numbers.

Answer 1

Correct Answer: 220

Step 1: Expression for $P_k$.
Perimeter of a square with side length $k$ is:
\[ P_k = 4k. \]
Step 2: Write the required sum.
\[ P_1 + P_2 + \dots + P_{10} = 4(1) + 4(2) + \dots + 4(10). \]
Factor out 4:
\[ = 4(1 + 2 + \dots + 10). \]
Step 3: Use formula for sum of first $n$ natural numbers.
\[ 1 + 2 + \dots + n = \frac{n(n+1)}{2}. \]
For $n=10$:
\[ 1 + 2 + \dots + 10 = \frac{10(11)}{2} = 55. \]
Step 4: Final calculation.
\[ 4 \times 55 = 220. \]
Final Answer:
\[ \boxed{220}. \]