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The sum of the first 6 terms of an arithmetic progression (AP) is 42. The ratio of its 10ᵗʰ term to the 38ᵗʰ term is 1:3. Calculate the first and the 13ᵗʰ term of the AP.

Answer :

Final answer:

To solve the problem, we make use of the formula for the sum of an AP series and the formula for the n-th term of an AP. We first solve for 'a' and 'd' using the sum condition and then confirm the obtained values with the term ratio condition. With known 'a' and 'd', we can find the first and 13th term of the AP.

Explanation:

The problem is about finding terms of an Arithmetic Progression (AP).

First, we know that the sum of first 6 term of the AP is 42. We use the formula for the sum of an AP series, S = n/2 * (2a + (n-1)d).

We replace S with 42, n with 6 and solve the equation to find 'a' and 'd' (first term and common difference).

Next, we know that the ratio of 10ᵗʰ term to 38ᵗʰ term is 1:3.

We use the formula for the nᵗʰ term of an AP, a + (n-1)*d, for both terms and solve to validate our previously found 'a' and 'd'.

With calculated value of 'a' and 'd', we can determine any term of the AP.

Thus, we apply these values to find the first term (a) and the 13ᵗʰ term (a + 12d) of the AP.

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