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------------------------------------------------ The 4th term of an arithmetic progression (AP) is 7, and the 21st term is 41. Find:

(a) The common difference.

(b) The 1st term.

(c) The sum of the first 20 terms of the AP.

To solve this problem, we need to recall the formula for the nth term of an AP (Arithmetic Progression), which is expressed as an = a + (n - 1)d, where an is the nth term, a is the first term, d is the common difference, and n is the number of terms.

Using the information given that the 4th term of an AP is 7 and the 21st term is 41:

4th term: a + 3d = 7
21st term: a + 20d = 41

Subtracting the first equation from the second gives us:

17d = 34 → d = 2

So, the common difference (b) is 2. Now, substituting for d, we find the first term:

a + 3(2) = 7 → a = 1

Thus, the first term (a) of the AP is 1.

To find the sum of the first 20 terms (S20), we use the formula: Sn = n/2 [2a + (n - 1)d]. Substituting the values:

S20 = 20/2 [2(1) + (20 - 1)(2)]

S20 = 10 [2 + 38]

S20 = 10 [40] = 400

Therefore, the sum of the first 20 terms of the AP is 400.

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