Question

The sum to $200$ terms of the series $1+4+6+5+11+6+.....$ is

• A. $29,600$
• B. $29,800$
• C. $30,200$
• D. None of these

Answer 1

The correct option is C $30,200$

Above series is a combination of two APs. The 1st AP is (1+6+11+.....) and the 2nd AP is (4+5+6+.....)
Since the terms of the two series alternate, $S=(1+6+11+....to100terms)$+(4+5+6+..... to 100 terms)
$=\frac{100[2\times 1+99\times 5]}{2}+\frac{100[2\times 4+99\times 1]}{2}$
(using the formula for the sum of an AP)
$=50[497+107]=50\left[604\right]=30200$
Alternatively, we can treat two consecutive terms as one. So we will have a total of 100 terms of the nature:
$(1+4)+(6+5)+(11+6)....\Rightarrow 5,11,17,.....$
Now, $a=5,d=6$ and $n=100$
Hence the sum of the given series is
$S=\frac{100}{2}\times [2\times 5+99\times 6]=50\left[604\right]=30200$