Gujarati
8. Sequences and Series
normal

The number of $5 -$tuples $(a, b, c, d, e)$ of positive integers such that

$I.$ $a, b, c, d, e$ are the measures of angles of a convex pentagon in degrees

$II$. $a \leq b \leq c \leq d \leq e$

$III.$ $a, b, c, d, e$ are in arithmetic progression is

A

$35$

B

$36$

C

$37$

D

$126$

(KVPY-2017)

Solution

(b)

We have,

$I$. $a, b,, c, d, e$ are angle of convex pentagon in degree.

$II$. $a \leq b \leq c \leq d \leq e$

$IIl$. $a, b, c, d, e$ are in $AP$

$a + b + c + d + e=540^{\circ}$

Let $a=\alpha$, common difference $=D$

$\therefore \quad \frac{5}{2}(2 a+4 D)=540^{\circ}$

$a+2 D=108$ and $a+4 D<180^{\circ}$

$[\because$ interior angle of polygon is

less than $\left.180^{\circ}\right]$

$\therefore \quad 108^{\circ}-2 D+4 D < 180^{\circ}$

$2 D < 180^{\circ}-108^{\circ}$

$0 < D < 36$

$\therefore$ Total 36 types are possible.

Standard 11
Mathematics

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