Hiroki and Mapiya were asked to find an explicit formula for the sequence 125\,,\,25\,,\,5\,,\,1,...125,25,5, 1,...125, comma, 25, comma, 5, comma, 1, comma, point, point, point. Hiroki said the formula is f(n)=625\cdot\left(\dfrac{1}{5}\rig ht)^{\large{n}}f(n)=625⋅( 5 1 ) n f, left parenthesis, n, right parenthesis, equals, 625, dot, left parenthesis, start fraction, 1, divided by, 5, end fraction, right parenthesis, start superscript, n, end superscript, and Mapiya said the formula is f(n)=125\cdot\left(\dfrac{1}{5}\rig ht)^{\large{n}}f(n)=125⋅( 5 1 ) n f, left parenthesis, n, right parenthesis, equals, 125, dot, left parenthesis, start fraction, 1, divided by, 5, end fraction, right parenthesis, start superscript, n, end superscript. Which one of them is right?

Mapiya's explicit formula : is right

Step-by-step explanation:

Sequence : 125,25,5,1,..

Hiroki's explicit formula :

Mapiya's explicit formula :

Formula of sequence=   a(r)^n

a = First term

n = No. of terms

r=Common ratio

a=125

So,

Hence Mapiya's explicit formula : is right

3. parallelogram

step-by-step explanation:

we turn each of our sides into direction vectors

ab=b-a=(3 - 7, 1 - 0) = (-4, 1)

bc=c-b=(1,6)

cd=d-c=(4,-1)

da=a-d=(-1,-6)

we see how the opposite sides have negated direction vectors; this is characteristic of a parallelogram.   we have congruent sides ab=cd and bc=da with unequal lengths because 1^2+6^2 is different from (-4)^2+1^2.   so it's not a square or a rhombus.  

we check the dot product of direction vectors of adjacent sides.   we have

ab dot bc = -4(1) + 1(6) = 2

that's not zero so the sides are not perpendicular.   we've ruled out rectangle.

the direction vectors indicate two pairs of congruent parallel sides, so we choose parallelogram over trapezoid.