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STD 11 - 8. sequence and series — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 11 - 8. sequence and series4 Marks
MCQ
If $x = \sum\limits_{n = 0}^\infty {{a^n}} ,\;y = \sum\limits_{n = 0}^\infty {{b^n},\;z = \sum\limits_{n = 0}^\infty {{{(ab)}^n}} } $, where $a,\;b < 1$, then
  • A
    $xyz = x + y + z$
  • ✓
    $xz + yz = xy + z$
  • C
    $xy + yz = xz + y$
  • D
    $xy + xz = yz + x$
✓

Answer

Correct option: B.
$xz + yz = xy + z$
b
(b) We have $x = \sum\limits_{n = 0}^\infty {{a^n}} = \frac{1}{{1 - a}}$

$\Rightarrow a = \frac{{x - 1}}{x}$

$y = \sum\limits_{n = 0}^\infty {{b^n}} = \frac{1}{{1 - b}}$

$ \Rightarrow $ $b = \frac{{y - 1}}{y}$

$z = \sum\limits_{n = 0}^\infty {{a^n}{b^n} = \frac{1}{{1 - ab}} \Rightarrow ab = \frac{{z - 1}}{z}} $

$\therefore $ $\frac{{x - 1}}{x}.\frac{{y - 1}}{y} = \frac{{z - 1}}{z}$

$ \Rightarrow $ $xy + z = zx + yz$.

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