Progression (Sequence and Series)HardBloom L1
Question
If in the arithmetic progression $\left\{ a_{n} \right\},a_{1} > 0,3a_{8} = 5a_{13}$, then among the following listed partial sums the largest one is
Options
A.$S_{10}$
B.$S_{11}$
C.$S_{20}$
D.$S_{21}$
Solution
Sol. [Key Concept: $3a_{8 = 5}a_{13}$ locates the last positive term; the partial sum is largest there.]
$$3\left( a_{1} + 7d \right) = 5\left( a_{1} + 12d \right) \Rightarrow 2a_{1} = - 39d$$
$$S_{n} = \frac{n}{2}\left( 2a_{1} + (n - 1)d \right) = \frac{n}{2}(n - 40),d < 0$$
$S_{n}$ is maximum at $n = 20$
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