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Sequence and Series

Multiple Choice Questions

71.

The interior angles of a polygon of n sides are in AP. The smallest angle is 120° and the common difference is 5°. What is the largest interior angle of the polygon ?

160 $°$ only
195 $°$ only
either 160 $°$ or 195 $°$
neither 160 $°$ nor 195 $°$

72.
$If x^{\ln (\frac{y}{z})} . x^{\ln {(xz)}^{2}} . z^{\ln (\frac{x}{y})} = y^{4 \ln (y)} for any x > 1, y > 1 and z > 1,$
then which one of the following is correct ?

ln(y) is the GM of ln(x), ln(x), ln(x) and ln(z)

ln(y) is the AM of ln(x), ln(x), ln(x) and ln(z)

ln(y) is the GM of ln(x), ln(x), ln(x) and ln(z)

ln(y) is the AM of ln(x), ln(x), ln(z) and ln(z)

ln(y) is the AM of ln(x), ln(x), ln(x) and ln(z)

$\ln {(a)}^{n} = nln (a) . . . (i) \ln (ab) = \ln (a) + \ln (b) \ln (\frac{a}{b}) = \ln (a) - \ln (b) \ln (x^{\ln (\frac{y}{z})} . y^{\ln {(xz)}^{2}} . z^{\ln (\frac{x}{y})}) = \ln (y^{4 ay}) \ln (x^{\ln (\frac{y}{z})}) + \ln (y^{\ln {(xz)}^{2}}) + \ln (z^{\ln (\frac{x}{y})}) = 4 \ln (y) (ay) \ln (\frac{y}{z}) \ln (x) + 2 \ln (xz) \ln (y) + \ln (\frac{x}{y}) \ln (z) = 4 {(\ln (y))}^{2} (\ln (y) - \ln (z)) \ln (x) + 2 (\ln (x) + \ln (z)) \ln (y) + [\ln (x) - \ln (y)] \ln (z) = 4 {(\ln (y))}^{2} \ln (y) \ln (x) - \ln (z) \ln (x) + 2 \ln (x) \ln (y) + \ln (x) \ln (z) - \ln (y) \ln (z) = 4 {(\ln (y))}^{2} 3 \ln (x) \ln (y) + \ln (z) \ln (y) = = 4 {(\ln (y))}^{2} \frac{\ln (x) + \ln (x) + \ln (x) + \ln (z)}{4} = \ln (y)$