The sum of the first 6 terms of an arithmetic progression (AP) is 42. The ratio of its 11th term to its 33rd term is 1:3. Calculate the first term of the AP.

Question

27-10-2024
Mathematics

High School

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The sum of the first 6 terms of an arithmetic progression (AP) is 42. The ratio of its 11th term to its 33rd term is 1:3. Calculate the first term of the AP.

Answer :

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Final answer:

To find the first term of the arithmetic progression, set up a system of equations using the given sum of the first six terms and the ratio of the 11th term to the 33rd term. Solve the system to find the first term, which is 7.

Explanation:

To calculate the first term of the arithmetic progression (AP), let's denote the first term as a and the common difference as d. The sum of the first six terms of an AP is given by the formula Sn = n/2 * (2a + (n-1)d). We know that S6 = 42, so we can write the equation 6/2 * (2a + 5d) = 42.

Next, we need to consider the ratio of the 11th term to the 33rd term. The nth term of an AP is given by Tn = a + (n-1)d. Therefore, the 11th term is a + 10d and the 33rd term is a + 32d. The ratio of these terms is 1:3, which means (a + 10d) / (a + 32d) = 1/3.

To find the first term, solve the system of two equations: 3(a + 10d) = a + 32d and 3 * (2a + 5d) = 42. By solving, you'll get two linear equations and can find the values of a and d. Through substitution and simplification, you derive that the first term, a, equals 7.