Five distinct positive integers are in …

Let a be the first term and d be the common positive difference .

Then,A.P is

a,a+d,a+2d,a+3d,a+4d

Sum of five distinct positive integers=10020

Sum of  nth term of A.P

{tex}S_n=\frac{n}{2}(a+a_n){/tex}

Where a=First term

a_n=nth term of A.P

n=Total number of terms in A.P

Using the formula

{tex}10020=\frac{5}{2}(a+a_n){/tex}

{tex}a+a_n=\frac{10020\times 2}{5}=4008{/tex}

nth term of A.P is given by

{tex}a_n=a+(n-1)d{/tex}

Where d=Common difference of A.P

Using the formula

{tex}a_5=a+4d{/tex}

Substitute the value

a+a+4d=4008

2(a+2d)=4008

{tex}a+2d=\frac{4008}{2}=2004{/tex}

a=2004-2d

Substitute the value of a

{tex}a_5=2004-2d+4d=2004+2d{/tex}

d=0 cannot be consider because d is positive

Therefore, possible value of d=1,2,3,..

Substitute d=1

{tex}a_5=2004+2(1)=2006{/tex}

for d=2

{tex}a_5=2004+2(2)=2008{/tex}

Hence, the smallest possible value of last term=2006