SUMMARY: The chapter on Arithmetic Progressions introduces the concept of sequences with a constant difference between consecutive terms and explores their properties and applications. KEY TOPICS: arithmetic progression, common difference, nth term formula, sum of n terms, derivation of formulas, applications of arithmetic progressions, solving problems, examples, exercises, real-life applications
What is the common difference in the arithmetic progression 5, 8, 11, 14?
A2
B3
C4
D5
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Correct answer: Option 1 — 2
Q21 Mark
If the first term of an arithmetic progression is 3 and the common difference is 7, what is the 10th term?
A66
B70
C73
D75
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Correct answer: Option 1 — 66
Q31 Mark
The sum of the first n terms of an arithmetic progression is given by the formula S_n = n/2 [2a + (n-1)d]. What does 'a' represent?
AThe last term
BThe first term
CThe common difference
DThe number of terms
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Correct answer: Option 2 — The first term
Q41 Mark
In an arithmetic progression, if the 5th term is 20 and the 10th term is 40, what is the common difference?
A4
B5
C6
D8
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Correct answer: Option 1 — 4
Q51 Mark
A person saves money in an arithmetic progression where the first month he saves $100 and increases his savings by $20 every month. How much will he save in the 12th month?
A$220
B$240
C$260
D$280
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Correct answer: Option 2 — $240
Short Answer Questions5 questions
Q63 Marks
Define an arithmetic progression and provide an example.
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An arithmetic progression (AP) is a sequence of numbers in which the difference between consecutive terms is constant. For example, the sequence 2, 5, 8, 11 is an AP with a common difference of 3.
Q73 Marks
What is the formula for the nth term of an arithmetic progression?
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The formula for the nth term (Tn) of an arithmetic progression is given by Tn = a + (n-1)d, where 'a' is the first term, 'd' is the common difference, and 'n' is the term number.
Q83 Marks
How do you find the sum of the first n terms of an arithmetic progression?
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The sum of the first n terms (Sn) of an arithmetic progression can be calculated using the formula Sn = n/2 * (2a + (n-1)d) or Sn = n/2 * (first term + last term).
Q93 Marks
If the first term of an arithmetic progression is 3 and the common difference is 4, what is the 10th term?
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Using the nth term formula Tn = a + (n-1)d, we can find the 10th term: T10 = 3 + (10-1) * 4 = 3 + 36 = 39.
Q103 Marks
A student finds that the sum of the first 15 terms of an AP is 210. If the first term is 5, what is the common difference?
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Using the sum formula Sn = n/2 * (2a + (n-1)d), we have 210 = 15/2 * (2*5 + (15-1)d). Solving this gives d = 2.
Long Answer Questions5 questions
Q116 Marks
Define an arithmetic progression (AP) and provide an example. How do you determine the common difference in an AP?
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An arithmetic progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is known as the common difference. For example, in the sequence 2, 5, 8, 11, the common difference is 3, as each term increases by 3 from the previous term. To determine the common difference, subtract any term from the term that follows it, such as 5 - 2 = 3.
Q126 Marks
Derive the formula for the nth term of an arithmetic progression. Explain each step of the derivation.
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The nth term (Tn) of an arithmetic progression can be derived from the first term (a) and the common difference (d). The formula is given by Tn = a + (n - 1)d. To derive this, we start with the first term a. The second term is a + d, the third term is a + 2d, and so on. Thus, the nth term can be expressed as a plus the common difference multiplied by (n - 1), which accounts for the number of intervals between the first term and the nth term.
Q136 Marks
Calculate the sum of the first 20 terms of the arithmetic progression where the first term is 5 and the common difference is 3. Show your calculations step by step.
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To calculate the sum of the first n terms (Sn) of an arithmetic progression, we use the formula Sn = n/2 * (2a + (n - 1)d). Here, a = 5, d = 3, and n = 20. First, we calculate 2a = 2 * 5 = 10. Next, we find (n - 1)d = 19 * 3 = 57. Now, we can substitute these values into the formula: Sn = 20/2 * (10 + 57) = 10 * 67 = 670. Therefore, the sum of the first 20 terms is 670.
Q146 Marks
Explain how arithmetic progressions can be applied in real-life situations. Provide at least two examples.
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Arithmetic progressions are widely used in various real-life situations. One example is in finance, where a person may save a fixed amount of money each month. If they start with an initial amount and add the same amount every month, the total savings form an arithmetic progression. Another example is in scheduling events, where a series of events occurs at regular intervals, such as a bus arriving every 15 minutes. The times at which the bus arrives can be represented as an arithmetic progression.
Q156 Marks
If the 7th term of an arithmetic progression is 20 and the common difference is 2, find the first term. Show your working.
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To find the first term (a) of the arithmetic progression, we use the formula for the nth term: Tn = a + (n - 1)d. Given that the 7th term (T7) is 20 and the common difference (d) is 2, we can substitute these values into the formula: 20 = a + (7 - 1) * 2. This simplifies to 20 = a + 12. To isolate a, we subtract 12 from both sides: a = 20 - 12 = 8. Therefore, the first term of the arithmetic progression is 8.
Assertion–Reason Questions5 questions
Q161 Mark
Assertion (A): An arithmetic progression is defined as a sequence where the difference between any two consecutive terms is constant.
Reason (R): The common difference in an arithmetic progression can be calculated by subtracting any term from the subsequent term.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q171 Mark
Assertion (A): The nth term of an arithmetic progression can be found using the formula a_n = a + (n-1)d.
Reason (R): This formula derives from the definition of arithmetic progression and allows the calculation of any term in the sequence.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q181 Mark
Assertion (A): The sum of the first n terms of an arithmetic progression can be calculated using the formula S_n = n/2 (2a + (n-1)d).
Reason (R): This formula is derived from the properties of arithmetic progressions and does not depend on the number of terms.
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Correct answer: Option 3 —
A is true, but R is false.
Q191 Mark
Assertion (A): If the common difference of an arithmetic progression is zero, then all terms of the sequence are equal.
Reason (R): This is true because a common difference of zero means that each term is the same as the previous term.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q201 Mark
Assertion (A): The sum of n terms of an arithmetic progression is always greater than zero.
Reason (R): The sum can be negative if the first term is negative and the common difference is also negative.
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Correct answer: Option 4 —
A is false, but R is true.
Statement-Based Questions5 questions
Q211 Mark
Statement 1: An arithmetic progression is defined as a sequence of numbers in which the difference between any two consecutive terms is always the same.
Statement 2: The common difference in an arithmetic progression can be negative, zero, or positive.
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Correct answer: Option 1 —
Both statements are true.
Q221 Mark
Statement 1: The nth term of an arithmetic progression can be calculated using the formula a_n = a + (n-1)d, where a is the first term and d is the common difference.
Statement 2: The sum of the first n terms of an arithmetic progression is given by S_n = n/2 (a + l), where l is the last term.
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Correct answer: Option 2 —
Only Statement 1 is true.
Q231 Mark
Statement 1: In an arithmetic progression, if the first term is 5 and the common difference is 3, the 10th term will be 32.
Statement 2: The sum of the first 10 terms of the arithmetic progression with first term 5 and common difference 3 is 155.
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Correct answer: Option 3 —
Only Statement 2 is true.
Q241 Mark
Statement 1: The sum of the first n terms of an arithmetic progression can also be expressed as S_n = n/2 (2a + (n-1)d).
Statement 2: An arithmetic progression cannot have all its terms as negative numbers.
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Correct answer: Option 2 —
Only Statement 1 is true.
Q251 Mark
Statement 1: If the first term of an arithmetic progression is 10 and the common difference is -2, the sequence will eventually become positive.
Statement 2: The common difference in an arithmetic progression determines whether the sequence is increasing or decreasing.
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Correct answer: Option 4 —
Both statements are false.
