Given the number series: 49, ?, 82, 145, 269, 484, 826. If the missing term is represented as $?$ and $? = A + 6$, find the nearest perfect square to the value of $A$.

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**Solution:**

1. The given series is: 49, ?, 82, 145, 269, 484, 826.

2. We need to find the missing term $?$ such that $? = A + 6$.

3. First, let's analyze the pattern in the series by finding the differences between consecutive terms (where possible):
   - Difference between 82 and ?: $82 - ?$
   - Difference between ? and 49: $? - 49$

4. Let's check the differences between known consecutive terms from 82 onwards:
   - $145 - 82 = 63$
   - $269 - 145 = 124$
   - $484 - 269 = 215$
   - $826 - 484 = 342$

5. Observe the pattern in these differences: 63, 124, 215, 342.
   - Differences between these differences:
     - $124 - 63 = 61$
     - $215 - 124 = 91$
     - $342 - 215 = 127$

6. The second-level differences are not consistent, so let's try another approach.

7. Alternatively, check if the series follows a pattern of adding increasing numbers:
   - From 49 to ?: $? - 49 = x$
   - From ? to 82: $82 - ? = y$

8. Suppose $x$ and $y$ follow a pattern similar to the later differences.

9. Let's try to find $?$ by assuming the difference between 49 and $?$ plus the difference between $?$ and 82 equals the difference between 82 and 145 (63):
   - $x + y = 63$

10. Also, since $?$ is between 49 and 82, $x$ and $y$ should be positive and satisfy:
    - $x = ? - 49$
    - $y = 82 - ?$

11. So, $x + y = ( ? - 49 ) + ( 82 - ? ) = 33$ which contradicts step 9.

12. Let's try another approach: check if the differences themselves form a pattern:
    - Differences: $? - 49$, $82 - ?$, $145 - 82 = 63$, $269 - 145 = 124$, $484 - 269 = 215$, $826 - 484 = 342$

13. The differences from 145 onwards are approximately increasing by around 60-70 each time.

14. Let's assume the difference between 49 and $?$ is $d_1$, and between $?$ and 82 is $d_2$.

15. If the differences increase approximately as 33, 49, 63, 124, 215, 342, then $d_1$ and $d_2$ might be 33 and 49.

16. So, $d_1 = ? - 49 = 33 ightarrow ? = 82$

17. But $?$ cannot be 82 because 82 is already in the series.

18. Let's try $d_1 = 33$, $d_2 = 33$:
    - $? = 49 + 33 = 82$ again same issue.

19. Try $d_1 = 27$, $d_2 = 35$:
    - $? = 49 + 27 = 76$
    - $82 - ? = 6$

20. This does not fit the increasing pattern.

21. Let's try to find $A$ by using the given relation $? = A + 6$ and check which $A$ fits best.

22. Try options for $A$ from the options given:
    - If $A=4$, then $?=10$, series: 49, 10, 82,... (10 is less than 49, unlikely)
    - If $A=8$, then $?=14$, series: 49, 14, 82,... (14 less than 49, unlikely)
    - If $A=5$, then $?=11$, series: 49, 11, 82,... (11 less than 49, unlikely)
    - If $A=11$, then $?=17$, series: 49, 17, 82,... (17 less than 49, unlikely)
    - If $A=7$, then $?=13$, series: 49, 13, 82,... (13 less than 49, unlikely)

23. All these values are less than 49, which is the first term, so the series is not strictly increasing.

24. Let's check the ratio of terms:
    - $49 	o ?$, $? 	o 82$, $82 	o 145$, etc.

25. Alternatively, check if the series follows a formula like $T_n = T_{n-1} + n^2$ or similar.

26. Calculate differences:
    - $82 - 49 = 33$
    - $145 - 82 = 63$
    - $269 - 145 = 124$
    - $484 - 269 = 215$
    - $826 - 484 = 342$

27. These differences are: 33, 63, 124, 215, 342.

28. Check if these differences correspond to squares or sums of squares:
    - 33 is close to $5^2 + 8 = 25 + 8$
    - 63 is close to $8^2 - 1 = 64 - 1$
    - 124 is close to $11^2 + 3 = 121 + 3$
    - 215 is close to $14^2 + 11 = 196 + 19$
    - 342 is close to $18^2 + 6 = 324 + 18$

29. This is inconclusive.

30. Since the problem states $? = A + 6$, and asks for the nearest square of $A$, and given the options, the best fit is $A = 5$ because 5 is a perfect square (nearest square to 5 is 4 or 9, but 5 itself is closest to 4).

**Answer:** $5$

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