Which of the following are squares of odd numbers? (i) 484 (ii) 961 (iii) 7396 (iv) 8649 (v) 4225. Without adding, find the sum: a) 1+3+5+7+9+11+13

Here are the solutions to the problems.

: Which of the following are squares of odd numbers?

To determine if a number is the square of an odd number, we find its square root and check if the root is an odd number.

• Step 1: Check (i) 484

  $$\sqrt{484} = 22$$

  Since 22 is an even number, 484 is the square of an even number.

• Step 2: Check (ii) 961

  $$\sqrt{961} = 31$$

  Since 31 is an odd number, 961 is the square of an odd number.

• Step 3: Check (iii) 7396

  $$\sqrt{7396} = 86$$

  Since 86 is an even number, 7396 is the square of an even number.

• Step 4: Check (iv) 8649 (assuming the partially visible number is 8649)

  $$\sqrt{8649} = 93$$

  Since 93 is an odd number, 8649 is the square of an odd number.

• Step 5: Check (v) 4225

  $$\sqrt{4225} = 65$$

  Since 65 is an odd number, 4225 is the square of an odd number.

The numbers that are squares of odd numbers are (ii) 961, (iv) 8649, and (v) 4225.

$$\boxed{\text{(ii) 961, (iv) 8649, (v) 4225}}$$

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: Without adding, find the sum:

We use the property that the sum of the first $n$ odd numbers is $n^2$.

• a) For (i) $1+3+5+7+9+11+13$

  • Step 1: Count the number of terms. There are 7 odd numbers in the sequence. So, $n=7$.
  • Step 2: Calculate the sum. The sum is $n^2 = 7^2 = 49$.
  $$\boxed{49}$$

• b) For (ii) $(1+3+5+7+9+11+13+1...)$ This expression is incomplete and cannot be solved accurately.

• c) For (iii) $1+3+5+7+9+11+13+15+17+19+21+23$

  • Step 1: Count the number of terms. There are 12 odd numbers in the sequence. So, $n=12$.
  • Step 2: Calculate the sum. The sum is $n^2 = 12^2 = 144$.
  $$\boxed{144}$$