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SpeechlessZzz9920
20.11.2020 •
Mathematics
(1) prove that: 1 + 4 + 7 +………+ (3n – 2) = (3n – 1). ( n ≥ 1 )
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Ответ:
The statement as given is not true. For n = 1, the left side is just the first term, 1, while the right side would be 3(1) - 1 = 2, and 1 ≠ 2.
I think what you meant to write was the equation
1 + 4 + 7 + ... + (3n - 2) = 1/2 n (3n - 1)
which can be easily proved by induction.
For n = 1, we have
1 = 1/2 (1) (3(1) - 1) → 1 = 1
which is of course true.
Now for the inductive step. Assume the equation holds for n = k, so that
1 + 4 + 7 + ... + (3k - 2) = 1/2 k (3k - 1)
and use this to show it holds for n = k + 1.
We would have
1 + 4 + 7 + ... + (3k - 2) + (3(k + 1) - 2)
which reduces to
1/2 k (3k - 1) + (3k + 1)
3/2 k² + 5/2 k + 1
1/2 (3k² + 5k + 2)
1/2 (k + 1) (3k + 2)
which is precisely what the formula gives on the right side for n = k + 1. QED
Ответ:
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Step-by-step explanation: i took the test