Prove that 3 + (3 times 5) + (3 times 5^2) + .. + (3 times 5^n) = (3 times (5^n

Question

Prove that 3 + (3 times 5) + (3 times 5^2) + .. + (3 times 5^n) = (3 times (5^n + 1 - 1)/4 whenever n is a nonnegative integer. a) Find a formula for 1/2 + 1/4 + 1/8 + .. + 1/2^n by examining the values of this expression for small values of n. b) Prove the formula you conjectured in part (a). Prove that for every positive integer n, (1 times 2) + (2 times 3) +. + (n times(n + 1)) = (n times(n + 1) times (n + 2))/3 Which amounts of money can be formed using just two-dollar bills and five-dollar bills? Prove your answer using strong induction. Find f(2), f (3), and f (4) if f is defined recursively by f(0) = -1, f(1) = 2, and for n = 1, 2, .. a) f(n + 1) = f(n) + 3f(n - 1) b) f(n + 1) = (f(n)^2) times (f(n - 1)) Assume f_n is the nth Fibonacci number. Prove that f_1 + f_3 +. + f_2n - 1 = f_2n when n is a positive integer

Explanation / Answer

1) Here is the given uestion:-

3+(3x5)+(3x5^2)+...+(3x5^n)= 3(5^n+1 - 1)/4

In order to prove it true, for n = 0:
LHS = 3,
while,
RHS = 3(5^1 - 1)/4
= 3(5-1)/4
= 3*3/4 = 3.
So, it's true for n = 0.

Now assume that it is true for n = k,
3 + 3(5) + 3(5^2) + ... + 3(5^k) = 3(5^(k+1) - 1)/4.

Now we Show it's also true for n = k + 1.

i.e.,

3 + 3(5) + 3(5^2) + ... + 3(5^k) + 3(5^(k+1)). (**) = 3(5^(k+2) - 1)/4.

Now as per the LHS (Left hand side):

By assumption, we know the 3 + 3(5) + 3(5^2) + ... + 3(5^k) part of (**) equals 3(5^(k+1) - 1)/4.

Thus, we use this:

3(5^(k+1) - 1)/4 + 3(5^(k+1))

by taking common denominator of 4 in the second half of the addition problem to get this:

3(5^(k+1) - 1)/4 + 12(5^(k+1))/4

add the numerators to get this:

[3(5^(k+1) - 1) + 12(5^(k+1))] / 4

=[3(5^(k+1)) - 3 + 12(5^(k+1))] / 4

=[15(5^(k+1)) - 3] / 4

Now, rewrite 15(5^(k+1)) as 3 * 5(5^(k+1)).
Since you now have an extra 5, this bumps up the power to k + 2. Now you have this:

[3(5^(k+2)) - 3] / 4

Factor out a 3 from the numerator to get this:

3(5^(k+2)-1)/4.

Hence Proved

Related Questions

Navigate

• Browse (All)
• Browse P
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.