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1. The 10th term of an arithmetic progression (AP) is given as 153.71.

Assignment:

1. The sum of the first 21 terms of the progression is 15.

2. The 10th and 15th terms of an AP are -5 and [tex]-7 \frac{1}{2}[/tex] respectively. What is the sum of the first 26 terms?

3. Find:
a) The tenth term.
b) The sum of the first 21 terms of the progression: [tex]-10, -8, -6, \ldots[/tex]

Answer :

Sure, let's go through the problems step-by-step!

### Problem 1:
Question: Find the 10th term of an arithmetic progression (AP) where the first term is 153 and the common difference is assumed to be -82.

To find the 10th term of an AP, we use the formula:
[tex]\[ a_{n} = a_1 + (n-1) \times d \][/tex]

- First term ([tex]\(a_1\)[/tex]) = 153
- Common difference ([tex]\(d\)[/tex]) = -82
- [tex]\(n\)[/tex] (term position) = 10

Substitute these values into the formula:
[tex]\[ a_{10} = 153 + (10-1) \times (-82) = 153 - 738 = -585 \][/tex]

### Problem 2:
Question: Find the sum of the first 21 terms of the AP.

The formula for the sum of the first [tex]\(n\)[/tex] terms of an AP is:
[tex]\[ S_n = \frac{n}{2} \times (2a_1 + (n-1) \times d) \][/tex]

- [tex]\(n\)[/tex] = 21
- Using the same first term ([tex]\(a_1\)[/tex]) = 153
- Same common difference ([tex]\(d\)[/tex]) = -82

Substitute these values into the formula:
[tex]\[ S_{21} = \frac{21}{2} \times (2 \times 153 + (21-1) \times -82) \][/tex]
[tex]\[ S_{21} = \frac{21}{2} \times (306 - 1640) \][/tex]
[tex]\[ S_{21} = \frac{21}{2} \times (-1334) \][/tex]
[tex]\[ S_{21} = -14007.0 \][/tex]

### Problem 3:
Question: If the 10th and 18th terms are -5 and [tex]\(-7.5\)[/tex] respectively, find the sum of the first 26 terms.

First, find the common difference ([tex]\(d\)[/tex]) where:
[tex]\[ a_{18} = a_{10} + (18-10) \times d \][/tex]
and
- [tex]\(a_{10} = -5\)[/tex]
- [tex]\(a_{18} = -7.5\)[/tex]

Re-arrange to find [tex]\(d\)[/tex]:
[tex]\[ -7.5 = -5 + 8d \][/tex]
[tex]\[ d = \frac{-2.5}{8} = -0.3125 \][/tex]

Next, calculate the first term [tex]\(a_1\)[/tex]:
[tex]\[ a_{10} = a_1 + (10-1) \times d \][/tex]
[tex]\[ -5 = a_1 + 9 \times -0.3125 \][/tex]
[tex]\[ a_1 = -5 + 2.8125 = -2.1875 \][/tex]

Using the sum formula, find the sum of the first 26 terms:
[tex]\[ S_{26} = \frac{26}{2} \times (2a_1 + (26-1) \times d) \][/tex]
[tex]\[ S_{26} = 13 \times (2 \times -2.1875 + 25 \times -0.3125) \][/tex]
[tex]\[ S_{26} = 13 \times (-4.375 - 7.8125) \][/tex]
[tex]\[ S_{26} = 13 \times (-12.1875) \][/tex]
[tex]\[ S_{26} = -158.4375 \][/tex]

### Problem 4:
Question: Find the 10th term and the sum of the first 21 terms for the sequence: [tex]\(-10, -8, -6, \cdots\)[/tex].

Part A: The 10th term

For this sequence, the first term ([tex]\(a_1\)[/tex]) is -10, and the common difference ([tex]\(d\)[/tex]) is 2. Using the formula for the 10th term:
[tex]\[ a_{10} = a_1 + (10-1) \times d \][/tex]
[tex]\[ a_{10} = -10 + 9 \times 2 = 8 \][/tex]

Part B: The sum of the first 21 terms

[tex]\[ S_{21} = \frac{21}{2} \times (2a_1 + (21-1) \times d) \][/tex]
[tex]\[ S_{21} = \frac{21}{2} \times (2 \times -10 + 20 \times 2) \][/tex]
[tex]\[ S_{21} = \frac{21}{2} \times (-20 + 40) \][/tex]
[tex]\[ S_{21} = \frac{21}{2} \times 20 \][/tex]
[tex]\[ S_{21} = 210.0 \][/tex]

I hope this helps! Let me know if you have any more questions!

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