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The first and last terms of an arithmetic progression are 1 and 121, respectively. Find the number of terms in the AP and the common difference if the sum of its terms is either 549 or 671.

Answer :

The common difference is 11. The number of terms in the AP is 9, and the common difference between the terms is 11.

The number of terms in the arithmetic progression (AP) can be found by using the formula:

Number of terms (n) = (Last term - First term) / Common difference + 1.

Given that the first term (a₁) is 1 and the last term (aₙ) is 121, we can substitute these values into the formula:

n = (121 - 1) / Common difference + 1.

To find the common difference, we need additional information. Let's proceed by using the sum of the AP's terms.

The sum of the terms in an AP can be calculated using the formula:

Sum (S) = (n/2) * (First term + Last term).

Given that the sum of the terms is 549, we can substitute the values into the formula:

549 = (n/2) * (1 + 121).

Simplifying the equation:

549 = (n/2) * 122.

Dividing both sides by 122:

4.5 = n/2.

Multiplying both sides by 2:

9 = n.

Therefore, the number of terms in the AP is 9.

To find the common difference, we can use the second sum given, which is 671.

671 = (9/2) * (1 + 121).

Simplifying the equation:

671 = (9/2) * 122.

Dividing both sides by 122:

5.5 = (9/2).

Multiplying both sides by 2:

11 = 9.

Therefore, the common difference is 11.

In summary, the number of terms in the AP is 9, and the common difference between the terms is 11.

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