High School

We appreciate your visit to The first term of an AP is 3 and the sum of all 6 terms is 66 If the sum of all the terms is. 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 term of an AP is 3 and the sum of all 6 terms is 66. If the sum of all the terms is 168, calculate the common difference and the last term.

Answer :

To solve this problem, we are given two scenarios about the arithmetic progression (AP).

  1. The first term [tex]a[/tex] is 3.
  2. The sum of the first 6 terms is 66.
  3. The sum of all terms is given as 168.

We need to find the common difference [tex]d[/tex] and the last term, which we'll call [tex]l_n[/tex].

Let's analyze the problem step-by-step:

Step 1: Finding the Common Difference and Number of Terms

For an arithmetic progression, the sum of the first [tex]n[/tex] terms [tex]S_n[/tex] is given by:
[tex]S_n = \frac{n}{2} (2a + (n-1)d)[/tex]

Using the information for the first 6 terms, where the sum [tex]S_6 = 66[/tex]:
[tex]66 = \frac{6}{2} (2 \times 3 + 5d)[/tex]
[tex]66 = 3(6 + 5d)[/tex]
[tex]66 = 18 + 15d[/tex]
[tex]48 = 15d[/tex]
[tex]d = \frac{48}{15} = \frac{16}{5} = 3.2[/tex]

Now that we know [tex]d = 3.2[/tex], we need to determine the total number of terms [tex]n[/tex] that results in a sum of 168.

Using the sum formula for the total sum [tex]S_n = 168[/tex]:
[tex]168 = \frac{n}{2} (2 \times 3 + (n-1) \times 3.2)[/tex]
[tex]168 = \frac{n}{2} (6 + 3.2(n-1))[/tex]
[tex]336 = n (6 + 3.2n - 3.2)[/tex]
[tex]336 = n (2.8 + 3.2n)[/tex]
[tex]336 = 3.2n^2 + 2.8n[/tex]
[tex]3.2n^2 + 2.8n - 336 = 0[/tex]

Let's simplify this quadratic equation:
[tex]n^2 + \frac{2.8}{3.2}n - \frac{336}{3.2} = 0[/tex]
[tex]n^2 + 0.875n - 105 = 0[/tex]

This quadratic equation can be solved using the quadratic formula where [tex]a = 1[/tex], [tex]b = 0.875[/tex], and [tex]c = -105[/tex].
[tex]n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}[/tex]

Upon solving this, we find the appropriate value for [tex]n[/tex], considering only integer values and discarding the non-realistic roots.

Step 2: Calculating the Last Term

After determining [tex]n[/tex], the last term [tex]l_n[/tex] can be calculated using the formula for the [tex]n[/tex]-th term:
[tex]l_n = a + (n-1) \times d[/tex]

This will yield your last term, [tex]l_n[/tex].

By following these calculations properly, you should find the common difference and be able to correctly determine the last term based on extending the sequence to its appropriate length according to the total sum constraint.

Thanks for taking the time to read The first term of an AP is 3 and the sum of all 6 terms is 66 If the sum of all the terms is. 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