Questions

1. Question 1

   Find the 8th term of the arithmetico-geometric sequence, where the arithmetic part is $1, 4, 7, \ldots$ and the geometric part is $5, 10, 20, \ldots$.

   SolutionTheory
2. Question 2

   Find the $n^{th}$ term of the arithmetico-geometric sequence, where the arithmetic part is $3, 7, 11, \ldots$ and the geometric part is $2, 6, 18, \ldots$.

   SolutionTheory
3. Question 3

   Consider the arithmetico-geometric sequence defined by arithmetic part: $a_{n+1} = 2n + 5$ and geometric part: $b_{n-2} = \dfrac{1}{9}(-3)^n$. Find the $n^{th}$ term and the sum of first three terms of the arithmetico-geometric sequence.

   SolutionTheory
4. Question 4

   Sum to $n$ terms the following series:

   (i)

   $1\cdot 2 + 3\cdot 4 + 5\cdot 8 + 7\cdot 16 + \cdots$

   (ii)

   $2\cdot 3 + 4\cdot 3^2 + 6\cdot 3^3 + 8\cdot 3^4 + \cdots$

   (iii)

   $2 + \dfrac{5}{4} + \dfrac{8}{4^2} + \dfrac{11}{4^3} + \cdots$

   (iv)

   $1 + \dfrac{3}{5} + \dfrac{5}{5^2} + \dfrac{7}{5^3} + \cdots$

   (v)

   $1 + \dfrac{4}{3} + \dfrac{7}{9} + \dfrac{10}{27} + \cdots$

   SolutionTheory
5. Question 5

   Sum the following infinite series:

   (i)

   $1 + \dfrac{3}{2} + \dfrac{5}{4} + \dfrac{7}{8} + \cdots$

   (ii)

   $2 + \dfrac{5}{3} + \dfrac{8}{9} + \dfrac{11}{27} + \cdots$

   SolutionTheory
6. Question 6

   Show that $2^{\frac{1}{2}} \cdot 4^{\frac{1}{4}} \cdot 8^{\frac{1}{8}} \cdot 16^{\frac{1}{16}} \cdots \infty = 4$.

   SolutionTheory
7. Question 7

   Show that $\sqrt{4} \cdot \sqrt[4]{16} \cdot \sqrt[8]{64} \cdot \sqrt[16]{256} \cdots \infty = 16$.

   SolutionTheory
8. Question 8

   Sum to $n$ terms the series $2 + 4x + 6x^2 + 8x^3 + \cdots$ where $x \neq 1$.

   SolutionTheory
9. Question 9

   Find the sum to $n$ terms of the series: $\dfrac{2n+1}{2n-1} + 3\left(\dfrac{2n+1}{2n-1}\right)^2 + 5\left(\dfrac{2n+1}{2n-1}\right)^3 + \cdots$.

   SolutionTheory
10. Question 10

    Prove that $1 + 2\left(1+\dfrac{1}{n}\right) + 3\left(1+\dfrac{1}{n}\right)^2 + \cdots$ to $n$ terms $= n^2$.

    SolutionTheory
11. Question 11

    Sum the series to $n$ terms $2 + 5x + 8x^2 + 11x^3 + \cdots$ and deduce the sum to infinity if $|x| < 1$.

    SolutionTheory