Question

The 10th term of an arithmetic progression is 35 and the 20th term is 65. What is the first term?

• A. 10
• B. 8
• C. 5
• D. 15

Hint:

Use the formula \(a_n = a + (n-1)d\) for arithmetic progression problems to find \(a\) and \(d\) by setting up simultaneous equations.

Answer 1

The Correct Option is B

Step 1: Let the first term be \(a\) and common difference be \(d\). 
The \(n\)th term of A.P. is given by: \[ a_n = a + (n-1)d. \] 
Step 2: Given, \[ a_{10} = a + 9d = 35, \] \[ a_{20} = a + 19d = 65. \] 
Step 3: Subtracting the first equation from the second: \[ (a + 19d) - (a + 9d) = 65 - 35 \implies 10d = 30 \implies d = 3. \] 
Step 4: Substitute \(d = 3\) into \(a + 9d = 35\): \[ a + 9 \times 3 = 35 \implies a + 27 = 35 \implies a = 8. \] 
Step 5: Check options, \(a=8\) is option (B).