we have to find** trace of matrix of order i , where 1<=i<=100.**

so from the question we can say that** ti follows a triplet** where.

if --->* imod3=0 : di =2i+1 [email protected]*

* imod3=1 : di=i+3 [email protected]*

* i mod3=2 : di=0 [email protected]*

**for i = { 3,6,9................................99 } we use @1 (33 terms)**

di={7,13,19.................................199} [email protected]

**for i= {1,4,7..................................100} we use @2 (34 terms)**

di={4,7,10,....................................103} [email protected]

**for i={2,5,8.................................98} we use @3 **

di = {0,0,0,0,0,...............................} [email protected]

therefore trace of matrix = **summation of di (1<=i<=100)**

=**using arithmetic progression sum eqn.**

= *(n/2) (first term +last term)*

**for eqn [email protected] : **

di={7,13,19.................................199} :** n=33, a=7 ,d=6, l=199**

**sum**= n/2 (a+l) = 33/2 (199+7) =**3399**

**for eqn @5 :**

di={4,7,10,....................................103} : **n=34 a=4 , d=3 ,l=103**

**sum **= n/2 (a+l) = 34/2 (4+103) =**1819**

**for eqn @6 :**

di = {0,0,0,0,0,...............................}

**sum =0**

### so total sum = 3399+1819+0 =** 5218 answer.**