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Question 1 / 4
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What is the next number in the series: 3, 11, 25, 45, ...?
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Solution: Step 1: Identify the pattern of differences between consecutive terms: 11 - 3 = 8, 25 - 11 = 14, 45 - 25 = 20. Step 2: Notice that the differences between terms are increasing: 8, 14, 20. Step 3: Determine the pattern of differences: The differences are increasing by 6 each time (14 - 8 = 6, 20 - 14 = 6). Step 4: Predict the next difference: 20 + 26 = 46. Step 5: Calculate the next term in the series: 45 + 26 = 71. Step 6: Verify that 71 is among the answer choices.
2
The ratio of two numbers A and B is 2:5. The sum of 6 terms of an arithmetic progression starting with A and a common difference of B - A is 285. Find B.
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Solution: Step 1: Express A and B using a common variable. Step 2: Find the sum of 6 terms of the arithmetic progression. Step 3: Solve for the variable and find B. Step 1: Let A = 2x, B = 5x. Step 2: Common difference = B - A = 3x. Step 3: Sum of 6 terms = 285 = 6/2 * (2*2x + (6-1)*3x). Step 4: Simplify: 285 = 3 * (4x + 15x) = 3 * 19x. Step 5: Solve for x: 285 = 57x, x = 5. Step 6: B = 5x = 5 * 5 = 25.
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In an arithmetic progression, the second term is 8 and the eighth term is 2 more than three times the second term. What is the sum of the first 8 terms?
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Solution: Step 1: Let the first term be 'a' and common difference be 'd'. Given that the second term, a + d = 8. Step 2: The eighth term is given as a + 7d = 2 + 3(a + d) = 2 + 3*8 = 26. Step 3: Now we have two equations: a + d = 8 and a + 7d = 26. Step 4: Solve for d: Subtract the first equation from the second to get 6d = 18, which gives d = 3. Step 5: Substitute d back into the first equation to find a: a + 3 = 8, so a = 5. Step 6: Calculate the sum of the first 8 terms using the formula: S_n = n/2 * [2a + (n-1)d]. Step 7: Substituting the values: S_8 = 8/2 * [2*5 + (8-1)*3] = 4 * [10 + 21] = 4 * 31 = 124.
4
Complete the series: 236, 230, X, 201, 174, 136. Find the value of X.
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Solution: Step 1: Identify the series pattern by finding differences between numbers. Step 2: Differences between consecutive numbers are 6, ?, 18, 27, 38. Step 3: Second differences are constant at 5, 7, 9, 11. Step 4: Assume X - 230 = 11 (continuing pattern of second differences). Step 5: Solve for X: X = 230 + 11 = 241, but this doesn't fit. Step 6: Re-evaluate differences and second differences. Step 7: Correct second differences: 6, 11, 18, 27, 38. Step 8: First differences: 6 (from 236 to 230), so next difference should continue pattern. Step 9: Correct calculation: 236 - 230 = 6, 230 - 219 = 11, 219 - 201 = 18, 201 - 174 = 27, 174 - 136 = 38. Step 10: Hence, X = 219.
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