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Answer :
Final answer:
To find the common difference of an arithmetic progression, you can use the formula a_n = a_1 + (n - 1)d. Using this formula, we can solve for the common difference of the AP in this question. So, the correct option is 'None of the above'.
Explanation:
To find the common difference of an arithmetic progression (AP), we can use the formula:
an = a1 + (n - 1)d
Where an represents the nth term, a1 represents the first term, n represents the term number, and d represents the common difference.
Using the given information, we know that the 5th term of the AP is 20, so a5 = 20. Substituting this value into the formula, we have:
20 = a1 + (5 - 1)d
a1 + 4d = 20
We also know that the sum of the 7th and 11th terms is 64, so a7 + a11 = 64. Substituting the formulas for the terms, we have:
a1 + 6d + a1 + 10d = 64
2a1 + 16d = 64
Simplifying both equations, we get:
a1 + 4d = 20 ......(1)
2a1 + 16d = 64 ......(2)
Multiplying equation (1) by 2, we get:
2(a1 + 4d) = 2(20)
2a1 + 8d = 40 ......(3)
Subtracting equation (3) from equation (2), we eliminate the variable a1 and solve for d:
(2a1 + 16d) - (2a1 + 8d) = 64 - 40
8d = 24
d = 3
Therefore, the common difference of the AP is 3.
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