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 mean of 12 observations is 75. If two observation are discarded, then the mean of the remaining observations is 65. What is the mean of the discarded observations?
The arithmetic mean of 1, 8, 27, 64, … up to n terms is given by
Let a, b, c be in AP and k ≠ 0 be a real number. Which of the following are correct?
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3. \(\frac{a}{k},\frac{b}{k},\frac{c}{k}\) are in AP
Select the correct answer using the code given below:How many two-digit numbers are divisible by 4?
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Important Questions from Arithmetic Progressions
The mean of 12 observations is 75. If two observation are discarded, then the mean of the remaining observations is 65. What is the mean of the discarded observations?
Calculate the value of x if the arithmetic mean of the following data is zero-
Numbers Frequency x + 3 3 x - 7 7 x - 4 11 The arithmetic mean of 1, 8, 27, 64, … up to n terms is given by
Let a, b, c be in AP and k ≠ 0 be a real number. Which of the following are correct?
1. ka, kb, kc are in AP
2. k - a, k - b, k - c are in AP
3. \(\frac{a}{k},\frac{b}{k},\frac{c}{k}\) are in AP
Select the correct answer using the code given below:How many two-digit numbers are divisible by 4?
