MP Board · Class 10 · Mathematics · Arithmetic ProgressionsIf the sum of the first 14 terms of an AP is 1050 and its first term is 10, find the 20th term. Show complete step-by-step working.

Given Data:

• Sum of first 14 terms ($S_{14}$) = 1050
• First term ($a$) = 10
• Number of terms ($n$) = 14

Formula:\nThe sum of the first $n$ terms of an AP is given by:

$$S_n = \frac{n}{2} [2a + (n - 1)d]$$

Step-by-Step Calculation:

1. Substitute the given values into the sum formula for $n = 14$: $$S_{14} = \frac{14}{2} [2(10) + (14 - 1)d]$$ $$1050 = 7 [20 + 13d]$$

2. Divide both sides by 7: $$\frac{1050}{7} = 20 + 13d$$ $$150 = 20 + 13d$$

3. Solve for $d$: $$150 - 20 = 13d$$ $$130 = 13d$$ $$d = \frac{130}{13}$| $$d = 10$$

4. Now, find the 20th term ($a_{20}$) using the $n$-th term formula: $$a_n = a + (n - 1)d$$ $$a_{20} = 10 + (20 - 1)(10)$$ $$a_{20} = 10 + (19)(10)$$ $$a_{20} = 10 + 190$$ $$a_{20} = 200$$

Final Answer:\nThe 20th term of the given AP is 200.