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Answer :
Final answer:
To identify the arithmetic progression and the number of terms, we establish equations based on the sum of terms and solve them to determine the first term, common difference, and number of terms.
Explanation:
To find the arithmetic progression, let's denote the first term as a and the common difference as d. Given that the sum of the 2nd and 4th terms is 22, we can write this as: a + d + a + 3d = 22. Simplifying this gives us 2a + 4d = 22.
The sum of the first 11 terms is 253, which can be stated using the formula for the sum of an arithmetic series: Sn = n/2[2a + (n-1)d]. Plugging in the values, we get: 11/2[2a + 10d] = 253.
From the two equations:
- 2a + 4d = 22
- 11/2[2a + 10d] = 253
We can solve for a and d. Then, to find the number of terms, knowing the last term is 67, we use the nth term of an AP formula: a + (n-1)d = 67.
After calculating the values of a and d, we plug them into the last equation to find n.
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