Questions

1. Question 1

   Use mathematical induction to prove the following formulae for every positive integer $n$.

   (i)

   $\log x^n = n \log x$, where $x$ is positive

   (ii)

   $2 + 5 + 8 + \cdots + (3n-1) = \dfrac{n}{2}(3n+1)$

   (iii)

   $2 + (2+5) + (2+5+8) + \cdots + \dfrac{n}{2}(3n+1) = \dfrac{n}{4}(n+1)^2$

   (iv)

   $2 + 6 + 18 + \cdots + 2 \times 3^{n-1} = 3^n - 1$

   (v)

   $1\times 3 + 2\times 5 + 3\times 7 + \cdots + n\times(2n+1) = \dfrac{n(n+1)(4n+5)}{6}$

   (vi)

   $\dfrac{1}{1\times 2} + \dfrac{1}{2\times 3} + \dfrac{1}{3\times 4} + \cdots + \dfrac{1}{n(n+1)} = 1 - \dfrac{1}{n+1}$

   (vii)

   $r + r^2 + r^3 + \cdots + r^n = \dfrac{r(1-r^n)}{1-r}$, $(r \neq 1)$

   (viii)

   $a + (a+d) + (a+2d) + \cdots + [a+(n-1)d] = \dfrac{n}{2}[2a+(n-1)d]$

   (ix)

   $a_n = a_1 + (n-1)d$ when $a_1, a_1+d, a_1+2d, \ldots$ form an A.P.

   (x)

   $a_n = a_1 r^{n-1}$ when $a_1, a_1 r, a_1 r^2, \ldots$ form a G.P.

   (xi)

   $\dbinom{3}{3} + \dbinom{4}{3} + \dbinom{5}{3} + \cdots + \dbinom{n+2}{3} = \dbinom{n+3}{4}$

   (xii)

   The sum of first $n$ odd natural numbers is $n^2$.

   SolutionTheory
2. Question 2

   Prove by mathematical induction that for all positive integral values of $n$

   (i)

   $n^2 + n$ is divisible by 2

   (ii)

   $5^n - 2^n$ is divisible by 3

   (iii)

   $8 \times 10^n - 2$ is divisible by 6

   SolutionTheory
3. Question 3

   Prove that $\sum_{k=1}^{n} r^k = \dfrac{r^{n+1}-1}{r-1}$, whenever $n$ is a positive integer.

   SolutionTheory
4. Question 4

   $x - y$ is a factor of $x^n - y^n$ for all positive integral values of $n$, $(x \neq y)$.

   SolutionTheory
5. Question 5

   $n! > 2^n - 1$ for integral values of $n \ge 4$.

   SolutionTheory
6. Question 6

   $4^n > 3^n + 2^{n-1}$ for integral values of $n \ge 2$.

   SolutionTheory
7. Question 7

   $1 + nx \le (1+x)^n$ for $n \ge 2$ and $x > -1$.

   SolutionTheory
8. Question 8

   Aliza invests Rs. 1,000,000 in a business that promises a 6% return compounded annually. Prove by mathematical induction that the amount of money after $n$ years is $1,000,000(1.06)^n$.

   SolutionTheory
9. Question 9

   A bank offers an investment with an annual interest rate $r$. If $P$ rupees are invested, the amount after $n$ years is given by: $A(n) = P(1+r)^n$. Prove by induction that this formula holds for all $n \ge 0$.

   SolutionTheory
10. Question 10

    Sikander saves Rs. 500 in the first month and increases his savings by Rs. 500 every subsequent month. Using mathematical induction, determine whether his total savings will reach at least Rs. 12,000 after 24 months.

    SolutionTheory
11. Question 11

    Prove by mathematical induction that if Ali takes a loan of Rs. 2,000,000 and pays Rs. 50,000 at the end of each year, the remaining balance after $n$ years is $R_n = 2,000,000 - 50,000n$.

    SolutionTheory
12. Question 12

    If Salman starts savings with Rs. 5,000 and saves an additional Rs. 1,000 at the end of every month, derive a formula $S(n)$ for his total savings after $n$ months. Prove the correctness of your formula using mathematical induction.

    SolutionTheory