Class 10 Mathematics Chapter 5 Arithmetic progressions

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a) 5, 9, 13, …

Solution Description: In an arithmetic progression (AP), each term increases by a fixed number called the common difference. We are given the 2nd term as 9 and the 5th term as 21. Since the terms follow a constant pattern, the difference between consecutive terms remains the same. To find this common difference, we observe the gap between the 2nd and 5th terms, which is three positions apart.

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a) Yes

Solution Description: To determine if the given sequence forms an arithmetic progression, we need to check whether the difference between consecutive terms is constant. In an arithmetic progression, the difference between each pair of consecutive terms remains the same. By examining the given sequence, we observe the difference between the first and second terms, the second and third terms, and the third and fourth terms. If all these differences are equal, then the sequence forms an arithmetic progression. In this case, the difference between each pair of consecutive terms is consistent, confirming that the sequence is indeed an arithmetic progression.

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c) 14, 17

Solution Description: The given sequence follows a pattern where each term increases by a fixed amount. Starting from 2, the next term is 5, then 8, and then 11. Observing the pattern, each term is obtained by adding the same value to the previous term. To find the next two terms, we continue this process by adding the same fixed difference to the last given term, 11. By following this pattern, we determine the next term and then repeat the process once more to get the following term.

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b) 59

Solution Description: To determine the 15th term of an arithmetic progression where the first term is 3 and the common difference is 4, we follow a systematic approach. An arithmetic progression follows a pattern where each term increases by a fixed amount. To find any specific term, we start with the first term and repeatedly add the common difference. Since we need the 15th term, we count how many times the common difference is added, which is one less than the term position.

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b) 8

Solution Description: To determine the first term of an arithmetic progression (AP), we analyze the given details. The 8th term of the sequence is 50, and the common difference between consecutive terms is 6. In an AP, each term is derived by repeatedly adding the common difference to the first term. Since the 8th term represents the result after adding the common difference seven times to the first term, we work backward by subtracting the common difference successively until we reach the first term. By tracing the sequence back step by step, we can determine the starting value of the progression.

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b) 2

[Solution Description]

We use the formula for the sum of the first n terms:

$$S_n = \frac{n}{2} [2a + (n-1)d]$$

Here, $S_{12} = 156$, $n = 12$, $d = 2$, and the first term a is unknown.

Substitute these values into the formula:

$$156 = \frac{12}{2} [2a + (12-1)\times2]$$

Simplify it:

$$156 = 6 [2a + 22]$$

Divide both sides by 6:

$$26 = 2a + 22$$

Solve for $a$:

$$2a = 26 – 22 = 4$$

$$a = \frac{4}{2} = 2$$

Therefore, the first term a is 2.

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c) 15

Solution Description: To determine how many terms must be taken in the arithmetic progression 18, 14, 10, …, so that their sum is -150, we start by identifying the pattern. This sequence has a common difference of -4, meaning each term decreases by 4. The sum of an arithmetic sequence depends on the number of terms, the first term, and the last term.

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c) 30

Solution Description: The given sequence follows a pattern where each term increases by a constant value. The first term is 3, the second term is 6, the third is 9, and the fourth is 12. Since the difference between consecutive terms remains the same, this sequence is an arithmetic progression. To find the sum of the first four terms, we simply add them together.

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b) 9th term

Solution Description: To determine which term in the arithmetic progression 3, 7, 11, 15 is equal to 35, we first identify the pattern in the sequence. The difference between consecutive terms is consistent, making it an arithmetic progression with a common difference. To find the term that equals 35, we need to establish its position in the sequence.

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b) 4500

Solution Description: In this savings scheme, the amount increases by a factor of 3/2 every 4 years. Given that the initial investment is 2000, the amount after 4 years will be 3/2 times the initial amount. After 8 years, which is two periods of 4 years, the amount will increase again by the same factor. By applying the growth factor twice, we can determine the final amount after 8 years. The solution involves multiplying the initial investment by the growth factor for each 4-year period, leading to the final amount after the total duration.

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d) 27

Solution Description: The given sequence starts with 7, followed by 12, 17, and 22. Observing the pattern, each term increases by a constant value. To determine the next term, we identify this consistent difference between consecutive terms. The difference between 12 and 7 is 5, between 17 and 12 is also 5, and the same pattern continues. Since this sequence follows a fixed pattern of addition, we can find the next term by adding this common difference to the last term, which is 22. By following this pattern, we can determine the next term in the sequence accurately.

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Solution Description: An arithmetic progression is a sequence in which the difference between consecutive terms remains constant. To determine which sequence is not an arithmetic progression, we check the differences between successive terms in each sequence.

For the first sequence (I), the difference between consecutive terms is always 5, showing a consistent pattern. Similarly, in the second sequence (II), the difference between consecutive terms is 6, which is also consistent.

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d) 32

Solution Description: The given sequence starts with 5, and each term increases by 3. This pattern of adding the same number each time indicates it is an arithmetic sequence. To find the 10th term, we count forward from the first term, adding 3 at each step. The second term is 8, the third is 11, the fourth is 14, and this pattern continues. By following this process up to the 10th term, we can determine its value.

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b) No, the diferences are not constant.

Solution Description: A sequence is an arithmetic progression (AP) if the difference between consecutive terms remains constant throughout. In the given sequence—10, 20, 30, and 45—we first examine the difference between successive terms. The difference between the first and second term is the same as that between the second and third, indicating a potential AP.