Question

prove (by induction) that (1+a)^{n} >1+na for all
nonzero real numbers a > -1 and all integers n>1

Answer #1

for the function g(x) = 1/x for all nonzero real
numbers X. Is the cardinality the same for all even numbers as it
is for all integers

Prove that if a is a transcendental number, then a^n is also
transcendental for all nonzero integers n.

1) Prove by induction that 1-1/2 + 1/3 -1/4 + ... - (-1)^n /n is always
positive
2) Prove by induction that for all positive integers n, (n^2+n+1)
is odd.

Using induction prove that for all positive integers n, n^2−n is
even.

Use Mathematical Induction to prove that 3 | (n^3 + 2n) for all
integers n = 0, 1, 2, ....

Using induction, prove the following:
i.) If a > -1 and n is a natural number, then (1 + a)^n >=
1 + na
ii.) If a and b are natural numbers, then a + b and ab are also
natural

Prove by induction on n that 13 | 2^4n+2 + 3^n+2 for all natural
numbers n.

Prove the following using induction:
(a) For all natural numbers n>2, 2n>2n+1
(b) For all positive integersn,
1^3+3^3+5^3+···+(2^n−1)^3=n^2(2n^2−1)
(c) For all positive natural numbers n,5/4·8^n+3^(3n−1) is
divisible by 19

Prove by induction that 5n + 12n – 1 is divisible by 16 for all
positive integers n.

If (xn) is a sequence of nonzero real numbers and if limn→∞ xn =
x where x does not equal zero; prove that lim n→∞ 1/ xn = 1/x

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