Observe the following pattern:

$2^{2} - 1^{2} = 2 + 1 3^{2} - 2^{2} = 3 + 2 4^{2} - 3^{2} = 4 + 3 5^{2} - 4^{2} = 5 + 4$

Find the value of
(i) $100^{2} - 99^{2}$ (ii) $111^{2} - 109^{2}$
(iii) $99^{2} - 96^{2}$

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Solution

From the given pattern,
$2^{2} - 1^{2} = 2 + 1 3^{2} - 2^{2} = 3 + 2 4^{2} - 3^{2} = 4 + 3 5^{2} - 4^{2} = 5 + 4$

Therefore
(i) $100^{2} - 99^{2} = 100 + 99 =$

(ii) $111^{2} - 109^{2} = 111^{2} - 110^{2} - 109^{2} = (111^{2} - 110^{2}) + (110^{2} - 109^{2}) = (111 + 110) + (110 - 109) = 221 + 219 = 440$

(iii) $99^{2} - 96^{2} = 99^{2} - 98^{2} + 98^{2} - 97^{2} + 97^{2} - 96^{2} = (99^{2} - 98^{2}) + (98^{2} - 97^{2}) + (97^{2} - 96^{2}) = (99 + 98) + (98 + 97) + (97 + 96) = 197 + 195 + 193 = 585$

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Similar questions

Q. Observe the following pattern
2² − 1² = 2 + 1
3² − 2² = 3 + 2
4² − 3² = 4 + 3
5² − 4² = 5 + 4
and find the value of
(i) 100² − 99²
(ii) 111² − 109²
(iii) 99² − 96²

Q. Observe the following pattern

1^{2} = \frac{1}{6} [1 \times (1 + 1) \times (2 \times 1 + 1)] 1^{2} + 2^{2} = \frac{1}{6} [2 \times (2 + 1) \times (2 \times 2 + 1)] 1^{2} + 2^{2} + 3^{2} = \frac{1}{6} [3 \times (3 + 1) \times (2 \times 3 + 1)] 1^{2} + 2^{2} + 3^{2} + 4^{2} = \frac{1}{6} [4 \times (4 + 1) \times (2 \times 4 + 1)]

and find the values of each of the following:
(i) 1² + 2² + 3² + 4²+ ... + 10²
(ii) 5² + 6² + 7² + 8² + 9² + 10² + 11² + 12²

Using the given pattern, find the missing numbers.
$1^{2} + 2^{2} + 2^{2} = 3^{2}$
$2^{2} + 3^{2} + 6^{2} = 7^{2}$
$3^{2} + 4^{2} + 12^{2} = 13^{2}$
$4^{2} + 5^{2} + . . .^{2} = 21^{2}$
$5^{2} + . . .^{2} + 30^{2} = 31^{2}$
$6^{2} + 7^{2} + . . . .^{2} = . . . .^{2}$ .

Q. The series

1^{2} - 2^{2} + 3^{2} - 4^{2} + . . . . . + 99^{2} - 100^{2} =

_____

Q. The value of

1^{2} - 2^{2} + 3^{2} - 4^{2} + 5^{2} - 6^{2} + . . . . + 99^{2} - 100^{2}

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Observe the following pattern: 22−12=2+132−22=3+242−32=4+352−42=5+4 Find the value of (i) 1002−992 (ii) 1112−1092 (iii) 992−962

Observe the following pattern:

$2^{2} - 1^{2} = 2 + 1 3^{2} - 2^{2} = 3 + 2 4^{2} - 3^{2} = 4 + 3 5^{2} - 4^{2} = 5 + 4$

Find the value of
(i) $100^{2} - 99^{2}$ (ii) $111^{2} - 109^{2}$
(iii) $99^{2} - 96^{2}$