Sum of First 30 Terms of an AP

Question

If the sum of the first $11$ terms of an arithmetic progression equals that of the first $19$ terms, then what is the sum of the first $30$ terms?

Accepted Answer

A. Let the AP have first term $a$ and common difference $d$. $$S_{11} = S_{19} \implies \frac{11}{2}[2a + 10d] = \frac{19}{2}[2a + 18d]$$ $$22a + 110d = 38a + 342d \implies 16a + 232d = 0 \implies 2a + 29d = 0$$ The sum of the first 30 terms is: $$S_{30} = \frac{30}{2}[2a + 29d] = 15 	imes (0) = 0$$

Answer

$0$

Answer

$-1$

Answer

$1$

Answer

Not unique

Sum of First 30 Terms of an AP

Explanation

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