ection 3.1 Homework1. 1+23 +... + n° = n2 (n + 1) for all natural numbers n.1II- +I+иfor all natural numbers n.I2.1(2) (2)3 3(4)n(n + 1) (n+1)1+ud-Ifor any r 1 and any neNи3. Show that k0=- I4. 1+2+22+ ... + 2"- = 2" -1 for all natural numbers n.52-1 is a multiple of 8 for all natural numbers n

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Asked Nov 24, 2019
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ection 3.1 Homework
1. 1+23 +... + n° = n2 (n + 1) for all natural numbers n.
1
I
I
- +
I
+
и
for all natural numbers n.
I
2.
1(2) (2)3 3(4)
n(n + 1) (n+1)
1+ud-I
for any r 1 and any neN
и
3. Show that k
0=
- I
4. 1+2+22+ ... + 2"- = 2" -1 for all natural numbers n.
52-1 is a multiple of 8 for all natural numbers n
help_outline

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ection 3.1 Homework 1. 1+23 +... + n° = n2 (n + 1) for all natural numbers n. 1 I I - + I + и for all natural numbers n. I 2. 1(2) (2)3 3(4) n(n + 1) (n+1) 1+ud-I for any r 1 and any neN и 3. Show that k 0= - I 4. 1+2+22+ ... + 2"- = 2" -1 for all natural numbers n. 52-1 is a multiple of 8 for all natural numbers n

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Expert Answer

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Step 1
1
Prove that
1
1
1
п
is true for all-natural numbers n
1(2) 2(3) 3(4)
п(п+1) п+1
Let's prove this by mathematical induction
Let n 1. Then
1
1
1(2)
+1
1
1(2) 2
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1 Prove that 1 1 1 п is true for all-natural numbers n 1(2) 2(3) 3(4) п(п+1) п+1 Let's prove this by mathematical induction Let n 1. Then 1 1 1(2) +1 1 1(2) 2

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Step 2
Thus, the statement is true for n 1
Assume that the statement is true for n k. Then,
k
1
1
1
1
1(2) 2(3) 3(4)
k(k+1)k1
Now, prove that the statement is true n = k+ 1
+
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Thus, the statement is true for n 1 Assume that the statement is true for n k. Then, k 1 1 1 1 1(2) 2(3) 3(4) k(k+1)k1 Now, prove that the statement is true n = k+ 1 +

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Step 3
1
1
+
+
1
1
1
k+1
2(3) 3(4)
(k+1) k+1((k+1) +1)
k+1+1
1
Substitute
1
in the above equation and simplify
1(2) 2(3) 3(4)
k(k 1 1
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1 1 + + 1 1 1 k+1 2(3) 3(4) (k+1) k+1((k+1) +1) k+1+1 1 Substitute 1 in the above equation and simplify 1(2) 2(3) 3(4) k(k 1 1

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