Determine the AP whose 5th term is 19 and \(a_{13}-a_{8}=20\).
\(a=3,\; d=4\) (AP: 3, 7, 11, …)
Let the first term of the AP be \(a\) and the common difference be \(d\).
Step 1: Write formula for nth term
The nth term of an AP is given by:
\(a_n = a + (n-1)d\).
Step 2: Use the condition for the 5th term
5th term = 19
\(a_5 = a + (5-1)d = a + 4d\).
So, \(a + 4d = 19\). (Equation 1)
Step 3: Use the condition \(a_{13} - a_8 = 20\)
13th term = \(a + 12d\)
8th term = \(a + 7d\)
Difference = \((a + 12d) - (a + 7d) = 5d\).
So, \(5d = 20\).
Therefore, \(d = 4\).
Step 4: Substitute value of d in Equation (1)
From Equation (1): \(a + 4d = 19\).
Put \(d = 4\):
\(a + 16 = 19\).
So, \(a = 3\).
Final Answer:
First term \(a = 3\), common difference \(d = 4\).
Hence, the AP is: 3, 7, 11, …