Given the two series:

Series 1: 87, B, (2a + 15), (302 + 6), (103 + Z), 1076, 1101

Series 2: (a + 58), 492, 529, Y, 639, 712, 749

Note: B = 87 + a

Find the value of B.

Question

Given the two series:

Series 1: 87, B, (2a + 15), (302 + 6), (103 + Z), 1076, 1101

Series 2: (a + 58), 492, 529, Y, 639, 712, 749

Note: B = 87 + a

Find the value of B.

Accepted Answer

**Solution:**

1. From the problem, we know that:
   - B = 87 + a

2. Look at Series 1: 87, B, (2a + 15), (302 + 6), (103 + Z), 1076, 1101
   - Simplify the known terms:
     - (302 + 6) = 308

3. Look at Series 2: (a + 58), 492, 529, Y, 639, 712, 749

4. We need to find the value of *a* first to find B.

5. Observe Series 2 for a pattern:
   - The numbers are: (a + 58), 492, 529, Y, 639, 712, 749
   - Calculate the differences between known consecutive terms:
     - 529 - 492 = 37
     - 639 - Y = unknown
     - 712 - 639 = 73
     - 749 - 712 = 37

6. Notice the differences 37 and 73 alternate or follow a pattern.

7. Check the difference between 492 and (a + 58):
   - 492 - (a + 58) = ?

8. Assume the difference between (a + 58) and 492 is 37 (like the last difference), then:
   - 492 - (a + 58) = 37
   - (a + 58) = 492 - 37 = 455
   - So, a = 455 - 58 = 397

9. Now calculate B:
   - B = 87 + a = 87 + 397 = 484

10. But 484 is not in the options. So check if the difference is 37 or 73.

11. Try difference 73:
    - 492 - (a + 58) = 73
    - (a + 58) = 492 - 73 = 419
    - a = 419 - 58 = 361

12. Then B = 87 + 361 = 448 (option d)

13. Check if this fits the pattern in Series 1:
    - B = 448
    - (2a + 15) = 2*361 + 15 = 722 + 15 = 737
    - (302 + 6) = 308
    - (103 + Z) unknown

14. The series 1 numbers are: 87, 448, 737, 308, (103 + Z), 1076, 1101
    - This seems inconsistent because 737 > 308, which is unusual in a series.

15. Try another approach: Since the problem is complex and only asks for B, and B = 87 + a, check the options for B and subtract 87 to get a:
    - Option (a) 548: a = 548 - 87 = 461
    - Option (b) 428: a = 428 - 87 = 341
    - Option (c) 568: a = 568 - 87 = 481
    - Option (d) 448: a = 448 - 87 = 361
    - Option (e) 528: a = 528 - 87 = 441

16. Among these, 548 (option a) gives a = 461.

17. Check if a = 461 fits Series 2:
    - (a + 58) = 461 + 58 = 519
    - Next term is 492, difference = 492 - 519 = -27 (negative, unlikely)

18. Check a = 428 - 87 = 341:
    - (a + 58) = 399
    - 492 - 399 = 93

19. Check a = 568 - 87 = 481:
    - (a + 58) = 539
    - 492 - 539 = -47 (negative)

20. Check a = 448 - 87 = 361:
    - (a + 58) = 419
    - 492 - 419 = 73

21. Check a = 528 - 87 = 441:
    - (a + 58) = 499
    - 492 - 499 = -7 (negative)

22. The only positive difference that fits a pattern is 73 (a=361), which corresponds to B = 448.

**Answer:** 448

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