High School

We appreciate your visit to The first third and ninth terms of an increasing arithmetic progression AP form the first three terms of a geometric progression G P If the. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!

The first, third, and ninth terms of an increasing arithmetic progression (AP) form the first three terms of a geometric progression (G.P.). If the first term of the AP is 3, find the common difference of the AP and the common ratio of the G.P.

Answer :

In this problem, we need to determine the common difference of an Arithmetic Progression (AP) and the common ratio of a Geometric Progression (G.P.). We're given that the first, third, and ninth terms of the AP form the first three terms of the G.P.

Here's a step-by-step solution:


  1. Identify the Terms of the AP:


    • The first term of the AP is given as [tex]a = 3[/tex].

    • Let's denote the common difference of the AP by [tex]d[/tex].

    • The terms in the AP are:

      • First term: [tex]a = 3[/tex]

      • Third term: [tex]a + 2d = 3 + 2d[/tex]

      • Ninth term: [tex]a + 8d = 3 + 8d[/tex]





  2. Set Up the Relationship for the G.P.:


    • Since these terms form a G.P., the ratio between consecutive terms must be equal.

    • Hence, [tex](a + 2d) = r \cdot a[/tex] and also [tex](a + 8d) = r \cdot (a + 2d)[/tex], where [tex]r[/tex] is the common ratio of the G.P.



  3. Express the Common Ratio [tex]r[/tex]:


    • From the first relationship:
      [tex]3 + 2d = r \cdot 3[/tex]
      [tex]r = \frac{3 + 2d}{3}[/tex]



  4. Substitute and Solve for [tex]d[/tex]:


    • Using the second relationship:
      [tex]3 + 8d = r \cdot (3 + 2d)[/tex]
      Substitute [tex]r[/tex] from the above equation:
      [tex]3 + 8d = \left( \frac{3 + 2d}{3} \right) (3 + 2d)[/tex]

    • Simplify:
      [tex]3 + 8d = \frac{(3 + 2d)^2}{3}[/tex]
      [tex]9 + 24d = (3 + 2d)^2[/tex]
      [tex]9 + 24d = 9 + 12d + 4d^2[/tex]
      [tex]4d^2 - 12d = 0[/tex]
      [tex]4d(d - 3) = 0[/tex]
      So, [tex]d = 0[/tex] or [tex]d = 3[/tex].



  5. Conclusion:


    • Since the AP is increasing, [tex]d = 3[/tex] is appropriate (as [tex]d = 0[/tex] would yield a constant sequence, which is not increasing).

    • Substitute [tex]d = 3[/tex] into the equation for [tex]r[/tex]:
      [tex]r = \frac{3 + 2(3)}{3} = \frac{9}{3} = 3[/tex]




Therefore, the common difference of the AP is [tex]d = 3[/tex] and the common ratio of the G.P. is [tex]r = 3[/tex].

Thanks for taking the time to read The first third and ninth terms of an increasing arithmetic progression AP form the first three terms of a geometric progression G P If the. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

Rewritten by : Barada