32. [tex]\((5x - 8), (x^2 - 2x)\)[/tex], and [tex]\((3x - 10)\)[/tex] are the third, fifth, and seventh terms respectively of an A.P.

(a) Find the value of [tex]\(x\)[/tex].
(b) Find the common difference and the first term of the series.
(c) Calculate the sum of the first 20 terms of the series.

(10 marks)



Answer :

Let's break this down and solve each part step-by-step.

### Given:
- The third term [tex]\((a_3)\)[/tex] = [tex]\(5x - 8\)[/tex]
- The fifth term [tex]\((a_5)\)[/tex] = [tex]\(x^2 - 2x\)[/tex]
- The seventh term [tex]\((a_7)\)[/tex] = [tex]\(3x - 10\)[/tex]

### Part (a) Find the value of [tex]\(x\)[/tex].

In an arithmetic sequence, the terms can be expressed in terms of the first term [tex]\(a_1\)[/tex] and the common difference [tex]\(d\)[/tex]:
- [tex]\(a_3 = a_1 + 2d\)[/tex]
- [tex]\(a_5 = a_1 + 4d\)[/tex]
- [tex]\(a_7 = a_1 + 6d\)[/tex]

Given:
[tex]\[ a_3 = 5x - 8 \][/tex]
[tex]\[ a_5 = x^2 - 2x \][/tex]
[tex]\[ a_7 = 3x - 10 \][/tex]

From the sequence we know:
[tex]\[ a_5 = a_3 + 2d \][/tex]

Substitute [tex]\(a_3\)[/tex]:
[tex]\[ x^2 - 2x = (5x - 8) + 2d \][/tex]

Solving for [tex]\(d\)[/tex]:
[tex]\[ x^2 - 2x - 5x + 8 = 2d \][/tex]
[tex]\[ x^2 - 7x + 8 = 2d \][/tex]
[tex]\[ d = \frac{x^2 - 7x + 8}{2} \][/tex]

Next, from:

[tex]\[ a_7 = a_3 + 4d \][/tex]

Substitute [tex]\(a_3\)[/tex]:
[tex]\[ 3x - 10 = (5x - 8) + 4d \][/tex]

Solving for [tex]\(d\)[/tex]:
[tex]\[ 3x - 10 - 5x + 8 = 4d \][/tex]
[tex]\[ -2x - 2 = 4d \][/tex]
[tex]\[ d = \frac{-2x - 2}{4} \][/tex]
[tex]\[ d = \frac{-x - 1}{2} \][/tex]

Set the expressions for [tex]\(d\)[/tex] equal to each other:
[tex]\[ \frac{x^2 - 7x + 8}{2} = \frac{-x - 1}{2} \][/tex]

Solving for [tex]\(x\)[/tex]:
[tex]\[ x^2 - 7x + 8 = -x - 1 \][/tex]
[tex]\[ x^2 - 6x + 9 = 0 \][/tex]
[tex]\[ (x - 3)^2 = 0 \][/tex]
[tex]\[ x = 3 \][/tex]

### Part (b) Find the common difference [tex]\(d\)[/tex] and the term of the series.

Substitute [tex]\(x = 3\)[/tex] back into the expression for [tex]\(d\)[/tex]:
[tex]\[ d = \frac{-3 - 1}{2} = \frac{-4}{2} = -2 \][/tex]

First term, [tex]\(a_1\)[/tex]:
[tex]\[ a_1 = a_3 - 2d \][/tex]
[tex]\[ a_3 = 5 \cdot 3 - 8 = 15 - 8 = 7 \][/tex]

[tex]\[ a_1 = 7 - 2(-2) \][/tex]
[tex]\[ a_1 = 7 + 4 = 11 \][/tex]

### Part (c) Calculate the sum of the first 20 terms of the series.

Sum of the first [tex]\(n\)[/tex] terms in an arithmetic series:
[tex]\[ S_n = \frac{n}{2} [2a_1 + (n-1)d] \][/tex]

Substitute [tex]\(n = 20\)[/tex], [tex]\(a_1 = 11\)[/tex], [tex]\(d = -2\)[/tex]:
[tex]\[ S_{20} = \frac{20}{2} [2 \cdot 11 + (20-1) \cdot (-2)] \][/tex]
[tex]\[ S_{20} = 10 [22 + 19 \cdot (-2)] \][/tex]
[tex]\[ S_{20} = 10 [22 - 38] \][/tex]
[tex]\[ S_{20} = 10 \cdot (-16) \][/tex]
[tex]\[ S_{20} = -160 \][/tex]

Thus, the sum of the first 20 terms is [tex]\(-160\)[/tex].

### Final Answers:
(a) The value of [tex]\(x\)[/tex] is [tex]\(3\)[/tex].

(b) The common difference [tex]\(d\)[/tex] is [tex]\(-2\)[/tex] and the first term [tex]\(a_1\)[/tex] is [tex]\(11\)[/tex].

(c) The sum of the first 20 terms is [tex]\(-160\)[/tex].