Observe the following pattern 22-12-2-132-22-3-242-32-4-352-42-5-4and find the value ofi 1002-992ii 1112-1092iii 992-962

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Byju's Answer

Standard X

Mathematics

Sequences, Series and Progressions

Observe the f...

Question

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

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Solution

From the pattern, we can say that the difference between the squares of two consecutive numbers is the sum of the numbers itself.
In a formula:

${(n + 1)}^{2} - {(n)}^{2} = (n + 1) + n$

Using this formula, we get:
(i) 100² − 99² = (99 + 1) + 99
= 199
(ii) 111² − 109² = 111² − 110²+ 110² − 109²
= (111 + 110) + (110 + 109)
= 440
(iii) 99² − 96² = 99² − 98²+ 98²− 97² + 97²− 96²
= 99 + 98 + 98 + 97 + 97 + 96
= 585

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Similar questions

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}

Q. The value of

100^{2} - 99^{2} + 98^{2} - 97^{2} . . . . . . . . . - 3^{2} + 2^{2} - 1^{2}

is:

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.
$1^{2} + 2^{2} + 2^{2} = 3^{2}$
$2^{2} + 3^{2} + 6^{2} = 7^{2}$
$3^{2} + 4^{2} + 12^{2} = 13^{2}$
$4^{2} + 5^{2} + . . .^{2} = 21^{2}$
$5^{2} + . . .^{2} + 30^{2} = 31^{2}$
$6^{2} + 7^{2} + . . . .^{2} = . . . .^{2}$ .

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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

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