Question

The sum of the first 20 terms of an AP is 820. If the first term is 5, find the 20th term and the common difference.

Hint:

Always use the formula \(S_n = \frac{n}{2}[2a + (n - 1)d]\) carefully and simplify each step to avoid errors in solving for \(d\).

Answer 1

Step 1: Use the formula for sum of first \(n\) terms of an AP: \[ S_n = \frac{n}{2}[2a + (n-1)d] \] Given: \[ S_{20} = 820, \quad a = 5, \quad n = 20 \] Step 2: Substitute the values: \[ 820 = \frac{20}{2}[2(5) + (20 - 1)d]
820 = 10[10 + 19d] \] Step 3: Solve for \(d\): \[ 10 + 19d = \frac{820}{10} = 82 \Rightarrow 19d = 72 \Rightarrow d = \frac{72}{19} \] Step 4: Find the 20th term: \[ a_{20} = a + 19d = 5 + 19 \cdot \frac{72}{19} = 5 + 72 = \boxed{77} \]