Step 1: Understanding the Sequence
The odd numbers from 1 to 100 form an arithmetic sequence: \[ 1, 3, 5, 7, \dots, 99 \] This sequence has:
First term \( a = 1 \)
Common difference \( d = 2 \)
Last term \( l = 99 \)
The number of terms (\( n \)) in this sequence is: \[ n = \frac{99 - 1}{2} + 1 = 50 \] Step 2: Sum of Squares Formula for Odd Numbers
The sum of squares of the first \( n \) odd numbers is given by: \[ S = \frac{n(2n+1)(2n-1)}{3} \] For \( n = 50 \): \[ S = \frac{50(101)(99)}{3} \] Step 3: Calculating the Sum \[ 50 \times 101 = 5050 \] \[ 5050 \times 99 = 499950 \] \[ S = \frac{499950}{3} = 166650 \]