Let a_1,,a_10 be an A.P.

Question

Let $a_1,\ldots,a_{10}$ be an A.P. with common difference $-3$, and $b_1,\ldots,b_{10}$ be a G.P. with common ratio $2$. Let $c_k=a_k+b_k$. If $c_2=12$ and $c_3=13$, then $\displaystyle\sum_{k=1}^{10}c_k$ is equal to \_\_\_\_\_\_\_\_.

Answer: 2021

Accepted Answer

$c_3-c_2 = (a_3-a_2)+(b_3-b_2) = -3+b_2 = 1 \Rightarrow b_2=4 \Rightarrow a_2=8.$

So the A.P. is $11,8,5,\ldots$ (first term $11$) and the G.P. is $2,4,8,\ldots$ (first term $2$).

$$\sum_{k=1}^{10}a_k = \frac{10}{2}\big[22+9(-3)\big] = 5(-5) = -25, \qquad \sum_{k=1}^{10}b_k = \frac{2(2^{10}-1)}{1} = 2046.$$

$$\sum c_k = -25+2046 = 2021.$$

Solutions · 1