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Answer :
Final answer:
To find the first and thirteenth term of an arithmetic progression (AP), we can use the given information about the sum of the first six terms, as well as the ratio of the 10th term to the 30th term. Using the formula for the sum of an AP and the relationship between consecutive terms, we can solve for the first term of the AP.
Explanation:
To find the first term of the AP, we can use the formula for the sum of an arithmetic progression:
Sum = n/2 * (first term + last term)
Where 'n' is the number of terms and 'first term' and 'last term' are the first and last terms of the AP, respectively. In this case, the sum of the first 6 terms is 42, so we have:
42 = 6/2 * (first term + last term)
Since this is an AP, we know that the difference between consecutive terms is always the same. So, we can express the last term in terms of the first term and the common difference 'd':
last term = first term + (n-1)d
Since the ratio of the 10th term to the 30th term is 1:3, we have:
10th term = first term + 9d
30th term = first term + 29d
Substituting these two new equations into the first one allows us to find a value for the first term:
42 = 6/2 * (first term + (first term + 29d))
Simplifying, we get:
42 = 3(first term + first term + 29d)
42 = 6(first term + 29d)
7 = 2(first term + 29d)
Solving for the first term, we find:
first term = 7/2 - 29d
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Final answer:
To find the first and thirteenth term of the arithmetic progression (AP), we need to use the given information. The sum of the first six terms of the AP is 42, and the ratio of the 10th term to the 30th term is 1:3. By using the formulas for the sum of an AP and the ratio of terms, we can solve for the first and thirteenth term.
Explanation:
To find the first and thirteenth term of the arithmetic progression (AP), we need to use the given information.
- Sum of the first six terms of the AP is 42
- Ratio of the 10th term to the 30th term is 1:3
Step 1: Find the common difference (d) of the AP. We know that the sum of the first six terms is 42, so we can use the formula: Sum = n/2 * (2a + (n-1)d) where n is the number of terms, a is the first term, and d is the common difference. Substitute the values: 42 = 6/2 * (2a + (6-1)d).
Step 2: Find the value of a (first term). We know that the ratio of the 10th term to the 30th term is 1:3. So, we can write: a + 9d = 1/3(a + 29d). Simplify and solve this equation to find the value of a.
Step 3: Calculate the first and thirteenth term of the AP. Once you have the value of a and the common difference (d), you can use the formulas: First term (a) = a and Thirteenth term = a + 12d to calculate the values.
