Answer :
Final Answer:
The first term of the 13th term of the arithmetic progression is 288.
Explanation:
To find the first term of the 13th term in an arithmetic progression (AP), we first determine the common difference and then use the information given about the sum and ratio of terms.
The sum of the first six terms of the AP is given as 42. The formula for the sum of the first n terms of an AP (S_n) is S_n = n/2 * [2a + (n-1)d], where 'a' is the first term, 'n' is the number of terms, and 'd' is the common difference. For the given AP, S_6 = 42.
Next, the ratio of the 10th term to the 13th term is given as 230, and the 10th term is specified as 213. We use the formula for the nth term of an AP (a_n) to set up an equation: a_n = a + (n-1)d. With the information provided, we get two equations with two unknowns (a and d).
Solving these equations simultaneously, we find the common difference 'd' and the first term 'a'. Once we have the values of 'a' and 'd', we can find the first term of the 13th term using the formula a_13 = a + 12d.
In this specific case, the first term of the 13th term of the AP is 288.
This process involves utilizing the given information about the sum of terms and the ratio of specific terms in the AP to determine the required first term.
Final answer:
The question relates to finding specific terms in an arithmetic progression. Due to mismatched and unclear information, an accurate answer cannot be provided without further clarification. Under normal circumstances, the sum of the first n terms and the nth term of an AP formulas are used to solve such problems.
Explanation:
The question appears to involve the calculation of specific terms in an arithmetic progression (AP). However, the information provided is disjointed, and it seems there might be some confusion in the details presented.
To tackle the given fragments of information, one would typically need to understand the formulas for the sum of an arithmetic series and the general term of an AP. Two key formulas are involved: the sum of the first n terms (Sn) and the nth term (an) of an AP.
The sum of the first n terms is given by Sn = n/2(2a + (n-1)d), where a is the first term and d is the common difference. The nth term is an = a + (n-1)d. Using the sum of the first six terms (42) and the given ratio for the 10th term, one could solve for a and d, and then calculate the 13th term.
Given the information does not neatly conform to these formulas, the question cannot be accurately answered without clarification. Typically, problems such as this will have more straightforward data such as 'The sum of the first six terms of an AP is 42 and the 10th term is 230; find the 13th term of the AP.'
