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Using the fact that the sum of the first odd numbers is , find the value of 21 + 23 + 25 + … + 39.

Class 8Squares and Square RootsSquares and Roots
  • A300
  • B400
  • C900
  • D500

Step-by-Step Solution

The odd numbers up to 39 are the first 20 odd numbers, with sum ; those up to 19 are the first 10, with sum . Subtract the unwanted start: 400 − 100 = 300. The trap 400 forgets to remove the first ten odd numbers, 900 squares the middle term 30, and 500 is a guess between the two squares.
Answer: A) 300

Key idea: Square root by long division

Use this for larger numbers (or non-perfect squares).

**Step 1**: Pair the digits from the right (e.g. 1849 → 18 49).
**Step 2**: Find the largest integer whose square ≤ first pair. Write it as the first quotient digit. Subtract its square.
**Step 3**: Bring down the next pair. Double the current quotient — that's your divisor's prefix. Find a digit such that (prefix × ≤ current dividend.
**Step 4**: Append to quotient, subtract, repeat.

Example: ≤ 18; remainder 2; bring down 49 → 249; double 4 = 8; find where ≤ 249; gives .) Quotient = 43.

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