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Devinlet n , n+1,n+2 be the three consecutive positive integers
n=1,2,3,4,5.......
let n be 1 :[1,1+1,1+2]
[1,2,3] the third no. only is divisible by 3
let n be 2:[2,3,4] only the second no. is divisible by 3
let n be 3: {3,4,5] only the first no. is divisible by 3
let n be 4: [4,5,6} only the third no. is divisible by 3
let n be 5:[5,6,7] only the second no. is divisible by 3
let n be 6: [6,7,8] only the first no. is divisible by 3
let n be 7: [7,8,9] only the third no. is divisible by 3
let n be 8: [8,9,10] only the second no. is divisible by 3
IN ALL THE CASES ONLY ONE OF ALL THE TRIPLETS IS DIVISIBLE BY 3
HENCE PROVED
this method is given in ncert examplar
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What is proof by induction?
Mathematical induction is just a way of proving a statement to be true for all positive integers: prove the statement to be true about 1; then assume it to be true for a generic integer x, and prove it to be true for x + 1; it therefore must be true for all positive integers.
Find two consecutive integers whose sum is 72?
There are no two consecutive integers that add up to 72. You can prove it this way: Let our numbers be represented by the variables "a" and "b". We are told: a = b + 1 a + b = 72 So we can use substitution to solve for either variable: (b + 1) + b = 72 2b + 1 = 72 2b = 71 b = 35.5 But 35.5 is not an integer, so this condition can not be met.
When you double the first and quadruple the second integer their sum is 30 find the consecutive integers?
There are no such consecutive integer as is so simple to prove! Suppose the first integer is x. Then the next (consecutive) integer is x+1. Then 2*x + 4*(x+1) = 30 So that 2x + 4x + 4 = 30 6x + 4 = 30 6x = 30 - 4 = 26 x = 26/6 which is NOT an integer.
How do you prove integers aren't closed under subtraction?
Take any two integers, and subtract one from another, you will have another integer. If there was a situation where you could show that this statement is not true, then that would prove your hypothesis, but I cannot think of any.
The sum of three concecutive integers is even true or false?
False. let the integers be n, n+1 and n+2 3n+3 is there sum and we need this to be even for all integers n. if n is odd, then 3n is odd ( take n=5 3x5=15 odd) any odd number +3 is even. if n is even, then 3n is even and an even number plus and 3 is odd so the answer is false You could just say or prove it is false with a single counter example. Take the 3 consecutive integers, 2,3,4 their sum is 9 and you are done. I mentioned the 3n+3 so you can see why it is false for even set of 3 when the first of the 3 consecutive numbers is even.
