Find the 31st term of an AP whose 11th term is 38 and the 16th term is 73.
178
Step 1: Use the general term of an AP.
For an arithmetic progression (AP), the \(n\)th term is given by
\[a_n = a + (n - 1)d\]
where \(a\) is the first term and \(d\) is the common difference.
Step 2: Translate the given information.
11th term is 38:
\[a_{11} = a + 10d = 38\]
16th term is 73:
\[a_{16} = a + 15d = 73\]
Step 3: Form equations and subtract.
From 11th term: \(a + 10d = 38\) ...(1)
From 16th term: \(a + 15d = 73\) ...(2)
Subtract (1) from (2):
\[(a + 15d) - (a + 10d) = 73 - 38\]
\[5d = 35\]
So,
\[d = \dfrac{35}{5} = 7\]
Step 4: Find the first term \(a\).
Use \(a + 10d = 38\):
\[a + 10 \cdot 7 = 38\]
\[a + 70 = 38\]
\[a = 38 - 70 = -32\]
Step 5: Find the 31st term.
\[a_{31} = a + 30d\]
Substitute \(a = -32\) and \(d = 7\):
\[a_{31} = -32 + 30 \cdot 7 = -32 + 210 = 178\]
Conclusion: The 31st term of the AP is 178.