If the sum of the first 14 terms of an AP is 1050 and its first term is 10, find the 20th term.
Step-by-Step Solution
Key Concept: Use the sum formula of an AP to determine the common difference, then apply the nth‑term formula to obtain the required term.
1. Write down the given data\
First term $a = 10$, number of terms $n = 14$, sum $S_{14}=1050$.
2. Use the sum formula of an AP\
$$S_n = \frac{n}{2}\bigl[2a + (n-1)d\bigr]$$
Substituting the known values:
$$1050 = \frac{14}{2}\bigl[2\times10 + (14-1)d\bigr]$$
$$1050 = 7\bigl[20 + 13d\bigr]$$
3. Solve for the common difference $d$\
Divide both sides by 7:
$$\frac{1050}{7} = 20 + 13d \quad\Rightarrow\quad 150 = 20 + 13d$$
$$13d = 150 - 20 = 130$$
$$d = \frac{130}{13} = 10$$
4. Find the 20th term using the nth‑term formula\
$$a_n = a + (n-1)d$$
For $n = 20$:
$$a_{20} = 10 + (20-1)\times10 = 10 + 19\times10 = 10 + 190 = 200$$
5. Answer\
The 20th term of the given AP is $\boxed{200}$.
Correct Answer: 200