The question "How many terms are in the AP 24 21 18?" is ambiguous. It seems to be asking for the number of terms in an arithmetic progression (AP) that starts with 24, 21, and 18, but it's missing crucial information like where the AP ends or what condition it must satisfy. Therefore, there are two possible interpretations:
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The question is incomplete: If the question refers to an AP that begins with 24, 21, and 18, but doesn't specify when the AP stops, or imposes some other condition, then we cannot determine a fixed number of terms.
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Implicit condition based on the reference: We can consider the provided reference to interpret the question as seeking the number of terms such that the sum of the terms is a particular value. The reference mentions "the number of terms of the A.P so that their sum is 78 are 4 and 13." Let's rephrase the question to fit the reference: "How many terms of the AP 24, 21, 18... must be taken so that the sum is 78?".
Solving for the Number of Terms with Sum = 78
Here's how to find the number of terms (n) such that the sum of the AP 24, 21, 18,... equals 78.
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Identify the AP properties:
- First term (a) = 24
- Common difference (d) = 21 - 24 = -3
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Use the formula for the sum of an AP:
Sn = n/2 * [2a + (n-1)d]
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Substitute the known values:
78 = n/2 * [2(24) + (n-1)(-3)]
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Simplify and solve for n:
156 = n [48 - 3n + 3]
156 = n [51 - 3n]
156 = 51n - 3n2
3n2 - 51n + 156 = 0
n2 - 17n + 52 = 0 -
Solve the quadratic equation:
(n - 4)(n - 13) = 0
Therefore, n = 4 or n = 13.
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Conclusion: So the number of terms of the A.P so that their sum is 78 are 4 and 13, which aligns with the reference.
Therefore, if we interpret the question as finding the number of terms such that the sum of the AP is 78, then the answer is 4 and 13.
