If 1 is added to the product of two consecutive even natural numbers

Discussion in 'Class 8 Maths Notes' started by sastry, Jan 28, 2011.

  1. sastry

    sastry New Member

    PATTERN 5 If 1 is added to the product of two consecutive even natural numbers, it is equal to the square of the only odd natural number between them.

    Explanation: We have,
    2x4÷1=9=32
    4x6÷1=25=52
    6 x8 +1= 49 = 72
    8x10÷1=81=92
    10x12÷1=121=112 etc.

    In general,

    2nx(2n +2) +1 =4n2 +4n +1 =(2n +1)2.

    ILLUSTRATIVE EXAMPLES

    Example 1 ‘The following numbes are not perfect squares. Give reason.
    (i) 1057 ii)-2’3453 (iii) 7928 (iv) 222222

    Solution We know that the natural numbers ending in the digits 2, 3, 7 or 8 are not perfect squares.
    Therefore, 1057, 23453, 7928 and 222222 are not perfect squares.

    Example 2 What will be the unit’s digit of the squares of the following numbers?
    (i) 71 (ii) 599 (iii) 2783 (iv) 1234

    Solution We know that the unit’s digit of the square of a number having digit at unit’s
    place as 1 or 9 is 1. Therefore, unit’s digit of the squares of numbers 71 and
    599 are each equal to 1. Since the unit’s digit of the square of a number having
    3 as unit’s digit is 9. So, the unit’s digit of the square of the number 2783 is 9. The unit’s digit of the square of the number 1234 is 6, because the unit’s digit of the square of a number having 4 or 6 at units place is 6.

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