Question

If every term of a G.P. with positive terms is the sum of its two previous terms, then the common ratio of the series is.

• A. 1
• B. $\frac { 2 } { \sqrt { 5 } }$
• C. $\frac { \sqrt { 5 } - 1 } { 2 }$
• D. $\frac { \sqrt { 5 } + 1 } { 2 }$

Answer 1

Answer: (D)

$\frac { \sqrt { 5 } + 1 } { 2 }$

Answer 2

Let first term and common ratio of G.P. are respectively $a$ and $r$, then under condition,

$T _ { n } = T _ { n - 1 } + T _ { n - 2 }$ $\Rightarrow$ $a r ^ { n - 1 } = a r ^ { n - 2 } + a r ^ { n - 3 }$

$\Rightarrow$ $a r ^ { n - 1 } = a r ^ { n - 1 } r ^ { - 1 } + a r ^ { n - 1 } r ^ { - 2 }$

$\Rightarrow$ $1 = \frac { 1 } { r } + \frac { 1 } { r ^ { 2 } }$ $\Rightarrow$ $r ^ { 2 } - r - 1 = 0$

$\Rightarrow$ $r = \frac { 1 \pm \sqrt { 1 + 4 } } { 2 } = \frac { 1 + \sqrt { 5 } } { 2 }$

Taking only (+) sign $( \because r > 1 )$ .