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------------------------------------------------ If the middle term of a finite arithmetic progression (AP) with 7 terms is 21, find the sum of all terms of the AP.

To find the sum of all terms of an AP with 7 terms where the middle term is 21, we can use the fact that the middle term is the fourth term in the progression. We calculate the sum as S₇ = 7/2 * 2 * 21, which equals 147.

The student asks about an arithmetic progression (AP) that has 7 terms, where the middle term is 21. To find the sum of all terms of the AP, we need to first determine the common difference and the first term of the progresssion. The middle term of an AP with an odd number of terms is also the median of the series, meaning it's exactly in the middle. Here, the middle (or fourth) term, a₄, can be represented as a₁ + 3d = 21.

Since we do not have the first term, a₁, or the common difference, d, we need another piece of information to solve for both. However, in an AP with 7 terms, the sum S₇ = n/2 * (a₁ + a₇), where a₇ is the last term. But we can also express the last term a₇ as a₁ + 6d. Thus, substituting the value of a₄ into the sum formula, we would have S₇ = 7/2 * (2a₄), since a₁ + 3d is essentially a₄.

The sum of all seven terms of the AP is thus S₇ = 7/2 * 2 * 21, which simplifies to S₇ = 7 * 21. Now, we can calculate the sum: S₇ = 147.