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heroxtree
05.03.2021 •
Mathematics
The number c is rational. Which statement about 7–c is true?
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Ответ:
Step-by-step explanation:
The given statement is:
6 + 12 + 18 + ... + 6n = 3n(n + 1)
we have t prove that the given statement is true for all positive integers n.
Thus, using mathematical induction, we have
Step 1: Let us show that the statement is true for n = 1.
Putting n=1 in the given statement, we have
6 = 3(1)(1 + 1). And this is clearly a true statement.
So the statement is true for n = 1.
Step 2: Show if it holds for k, then it holds for k + 1.
Let us assume that the statement is true for a positive integer k. That is,
6 + 12 + ... + 6k = 3k(k + 1).
Let us show that the statement must be true for the positive integer k + 1. We need to show that the sum of the first k + 1 multiples of 6 is equal to 3(k+1)( (k + 1) + 1).
Thus,
6 + 12 + ... + 6k + 6(k+1) = [6 + 12 + ... + 6k] + 6(k + 1)
= 3k(k + 1) + 6(k + 1) (this is true by the induction hypothesis)
= 3(k + 1) [ k + 2 ] (Factorizing 3(k + 1) )
= 3(k + 1)[(k + 1) + 1]
Therefore, the result holds.
Hence, the given statement is true for all positive integers n.