Let the Arithmetic Progression (AP) have the first term \(a\) and the common difference \(d\).
The formula for the \(n^{th}\) term of an AP is given by: \(a_n = a + (n-1)d\)
AP Term Calculation
We are given that the \(p^{th}\) term of the AP is \(k\). Using the formula:
\(a_p = a + (p-1)d = k \quad (1)\)Sum of Specific AP Terms
We need to find the sum of the \(p^{th}\) term, \((p+q)^{th}\) term, and \((p-q)^{th}\) term. Let's express these terms:
- \(p^{th}\) term: \(a_p = a + (p-1)d\)
- \((p+q)^{th}\) term: \(a_{p+q} = a + ((p+q)-1)d = a + (p+q-1)d\)
- \((p-q)^{th}\) term: \(a_{p-q} = a + ((p-q)-1)d = a + (p-q-1)d\)
The sum \(S\) is:
\(S = a_p + a_{p+q} + a_{p-q}\) \(S = [a + (p-1)d] + [a + (p+q-1)d] + [a + (p-q-1)d]\)Simplifying the Sum Expression
Combine like terms:
\(S = (a + a + a) + [(p-1)d + (p+q-1)d + (p-q-1)d]\) \(S = 3a + d[(p-1) + (p+q-1) + (p-q-1)]\)Simplify the expression inside the brackets:
\((p-1) + (p+q-1) + (p-q-1) = p - 1 + p + q - 1 + p - q - 1 = 3p - 3\)Substitute this back into the sum equation:
\(S = 3a + d(3p - 3)\) \(S = 3a + 3(p-1)d\)Final Calculation
Factor out 3:
\(S = 3[a + (p-1)d]\)From equation (1), we know that \(a + (p-1)d = k\). Substituting this value:
\(S = 3k\)Therefore, the sum of the \(p^{th}\) term, \((p+q)^{th}\) term, and \((p-q)^{th}\) term is \(3k\).
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?
