To replace the trees destroyed by typhoon Yolanda, the forestry department of Tagaytay has developed a ten-year plan. During the first year they will plant 100 trees. Each succeeding year, they will plant 50 more trees than they planted the year before.

a. 1,000 trees

b. 3,250 trees

Step-by-step explanation:

a. How many trees will they plant during the fifth year?

b. How many trees will they have planted by the end of the tenth year?

Using the sum of an arithmetic progression formula

Sn = n/2 {2a+(n-1)d}

Where,

Sn = Sum of an n terms

n = number of terms

a = first term

d = common difference

a.

n = 5

a = 100

d = 50

S5 = n/2 {2a+(n-1)d}

= 5/2 {2*100 + (5-1)50}

= 5/2 {200+(4)50}

= 5/2{200 + 200}

= 5/2(400)

= 2,000 / 2

= 1,000

S5 = 1,000 trees

b. Sn = n/2 {2a+(n-1)d}

n = 10

a = 100

d =50

S10 = 10/2{2*100 + (10-1)50}

= 5{200 + (9)50}

= 5{200 + 450}

= 5(650)

= 3,250

S10 = 3,250 trees

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.