The maximum sum of the series $$20 + 19\frac{1}{3} + 18\frac{2}{3} + 18 + .....\,{\text{is}}$$
A.
$$300$$
B.
$$310$$
C.
$$311\frac{2}{3}$$
D.
$$333\frac{1}{3}$$
Answer :
$$310$$
Solution :
The given series is arithmetic whose first term $$= 20,$$ and common difference $$ = - \frac{2}{3}$$
As the common difference is negative the terms will become negative after some stage. So the sum is maximum when all positive terms are added
Now, for the positive terms
$$\eqalign{
& {x_n} \geqslant 0 \cr
& \Rightarrow 20 + \left( {n - 1} \right) \times - \frac{2}{3} \geqslant 0 \cr
& \Rightarrow 60 - 2\left( {n - 1} \right) \geqslant 0 \cr
& \Rightarrow n \leqslant 31. \cr} $$
∴ The first 31 terms are non - negative
∴ Maximum sum
$$ = {S_{31}} = \frac{{31}}{2}\left[ {2 \times 20 + \left( {31 - 1} \right) \times - \frac{2}{3}} \right] = 310$$
Releted MCQ Question on Algebra >> Sequences and Series
Releted Question 1
If $$x, y$$ and $$z$$ are $${p^{{\text{th}}}},{q^{{\text{th}}}}\,{\text{and }}{r^{{\text{th}}}}$$ terms respectively of an A.P. and also of a G.P., then $${x^{y - z}}{y^{z - x}}{z^{x - y}}$$ is equal to:
If $$a, b, c$$ are in G.P., then the equations $$a{x^2} + 2bx + c = 0$$ and $$d{x^2} + 2ex + f = 0$$ have a common root if $$\frac{d}{a},\frac{e}{b},\frac{f}{c}$$ are in-