Arithmetic Progression: Finding the Term Number (m)
The problem asks for the value of m given the sum of the first n terms of an Arithmetic Progression (AP) and the value of the mth term.
Deriving the nth Term Formula
We are given the sum of the first n terms, \(S_n = 3n^2 + 5n\). The formula for the nth term (\(a_n\)) of an AP can be found using the relationship \(a_n = S_n - S_{n-1}\).
- Calculate \(S_{n-1}\): Replace n with (n-1) in the sum formula: \( S_{n-1} = 3(n-1)^2 + 5(n-1) \) \( S_{n-1} = 3(n^2 - 2n + 1) + 5n - 5 \) \( S_{n-1} = 3n^2 - 6n + 3 + 5n - 5 \) \( S_{n-1} = 3n^2 - n - 2 \)
- Calculate \(a_n\): Subtract \(S_{n-1}\) from \(S_n\): \( a_n = S_n - S_{n-1} \) \( a_n = (3n^2 + 5n) - (3n^2 - n - 2) \) \( a_n = 3n^2 + 5n - 3n^2 + n + 2 \) \( a_n = 6n + 2 \) This is the formula for the nth term of the AP.
Calculating the Value of m
We are given that the mth term, \(a_m\), is 68.
- Use the \(a_n\) formula for \(a_m\): Substitute m for n in the formula \(a_n = 6n + 2\): \( a_m = 6m + 2 \)
- Set \(a_m\) equal to the given value: \( 6m + 2 = 68 \)
- Solve for m: \( 6m = 68 - 2 \) \( 6m = 66 \) \( m = \frac{66}{6} \) \( m = 11 \)
Therefore, the value of m is 11.
Similar Questions
The first and the second terms of an AP are \(\frac{5}{2}\) and \(\frac{23}{12}\) respectively. If nth term is the largest negative term, what is the value of n ?
p, q, r and s are in AP such that p + s = 8 and qr = 15. What is the difference between largest and smallest numbers ?
In an AP, the first term is x and the sum of the first n terms is zero. What is the sum of next m terms ?
What is \(\displaystyle \sum_{n=1}^{34} a_n\) equal to ?
If the sum of the first 9 terms of an AP is equal to sum of the first 11 terms, then what is the sum of the first 20 terms ?
If the 5 th term of an AP is \(\frac{1}{10}\) and its 10 th term is \(\frac{1}{5},\) then what is the sum of first 50 terms ?
What is the arithmetic mean of 50 terms of an AP with first term 4 and common difference 4 ?
If x 2, x, -8 are in AP, then which one of the following is correct?
\(\frac{1}{b+c}, \frac{1}{c+a},\frac{1}{a+b}\) are in HP, then which of the following is/are correct?
1. a, b, c are in AP
2. (b + c) 2, (c + a) 2, (a + b) 2are in GP. Select the correct answer using the code given below.
If log 10 2, log 10 (2 x - 1), log 10 (2 x + 3) are in AP, then what is x equal to?
Important Questions from Arithmetic Progressions
The nth term of an A.P is \(\frac{{3 + {\rm{n}}}}{4}\) , then the sum of first 105 terms is
What is the sum of n terms of the series \(\sqrt 2 + \sqrt 8 + \sqrt {18} + \sqrt {32} + \ldots ?\)
Find the sum of all even numbers between 1 to 100.
The sum of $n$ terms of two arithmetic progressions are in the ratio $(9n + 5) : (5n + 21)$. Find the ratio of their $15^{th}$ terms.
Which of the following disciplines studies human populations mostly with respect to their size, their structure and their development?
