In an arithmetic progression where a₄ = 22 and a₉ = 57, what is a₁?
Answer and explanation
Correct answer: 1
For an arithmetic progression, aₙ = a₁ + (n − 1)d. Using the two given terms, a₉ − a₄ = [(a₁ + 8d) − (a₁ + 3d)] = 5d. Thus 57 − 22 = 35 = 5d, so d = 7. Now use the fourth term: a₄ = a₁ + 3d, giving 22 = a₁ + 21 and therefore a₁ = 1. Option A is correct. The other options do not satisfy both conditions simultaneously; for example, if a₁ were 7, the fourth term would be 28. The key idea is to find the common difference from the separation of the two known terms before finding the first term.
Frequently asked questions
What is the correct answer to this question?
1
Why is this the correct answer?
For an arithmetic progression, aₙ = a₁ + (n − 1)d. Using the two given terms, a₉ − a₄ = [(a₁ + 8d) − (a₁ + 3d)] = 5d. Thus 57 − 22 = 35 = 5d, so d = 7. Now use the fourth term: a₄ = a₁ + 3d, giving 22 = a₁ + 21 and therefore a₁ = 1. Option A is correct. The other options do not satisfy both conditions simultaneously; for example, if a₁ were 7, the fourth term would be 28. The key idea is to find the common difference from the separation of the two known terms before finding the first term.
Which subject and chapter does this question cover?
This is a Class 9 Mathematics question. Chapter: Sequences and Progressions. Topic: Arithmetic Progression.
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