The 26th, 11th and last terms are \(0,\;3\) and \(-\dfrac{1}{5}\), respectively. Find the common difference and number of terms.
\(d=-\dfrac{1}{5},\; n=27\)
Step 1: Write the formula for the \(n\)-th term of an AP.
In an arithmetic progression (AP), the \(n\)-th term is:
\(a_n = a + (n-1)d\)
where \(a\) is the first term, \(d\) is the common difference, and \(n\) is the term number.
Step 2: Use the information about the 26th term.
The 26th term is given as 0.
So, \(a + 25d = 0\) … (1)
Step 3: Use the information about the 11th term.
The 11th term is given as 3.
So, \(a + 10d = 3\) … (2)
Step 4: Subtract the two equations to find \(d\).
(1) – (2):
\((a + 25d) - (a + 10d) = 0 - 3\)
\(15d = -3\)
\(d = -\dfrac{1}{5}\)
Step 5: Find the first term \(a\).
Put \(d = -\dfrac{1}{5}\) in equation (2):
\(a + 10(-\dfrac{1}{5}) = 3\)
\(a - 2 = 3\)
\(a = 5\)
Step 6: Use the last term to find the number of terms.
Last term is \(-\dfrac{1}{5}\).
So, \(a + (n-1)d = -\dfrac{1}{5}\)
\(5 + (n-1)(-\dfrac{1}{5}) = -\dfrac{1}{5}\)
Multiply through by 5 to clear the fraction:
\(25 - (n-1) = -1\)
\(25 + 1 = n - 1\)
\(n - 1 = 26\)
\(n = 27\)
Final Answer: Common difference \(d = -\dfrac{1}{5}\), Number of terms \(n = 27\).