Priyanka and Ethan were asked to find an explicit formula for the sequence -3,-14,-25,-36,...−3,−14,−25,−36,.. .minus, 3, comma, minus, 14, comma, minus, 25, comma, minus, 36, comma, point, point, point, where the first term should be g(1)g(1)g, left parenthesis, 1, right parenthesis. Priyanka said the formula is g(n)=-3-11ng(n)=−3−11ng, left parenthesis, n, right parenthesis, equals, minus, 3, minus, 11, n. Ethan said the formula is g(n)=-3+11ng(n)=−3+11ng, left parenthesis, n, right parenthesis, equals, minus, 3, plus, 11, n. Which one of them is right?

step-by-step explanation:

part a. the differences are decreasing, so we know the sequence is not arithmetic. the ratios are constant (=3/4), so we know the sequence is geometric.

part b. from part a, we know the common ratio is 3/4, so each term is 3/4 of the one before. the initial term is the number of gallons at h=1.

g[h] = (3/4)·g[h-1]

g[1] = 384 . . . . initial value required for a recursive definition

part c. the general formula for a geometric sequence is

a[n] = a[1]·r^(n-1)

we have a[1] = 384, r = 3/4, and the term number given by h. the formula that follows is

g[h] = 384·(3/4)^(h-1)

part d. filling in the value and doing the arithmetic, we have

g[12] = 384·(3/4)^11 ≈ 384·0.042235136 ≈ 16.2183 . . gallons

part e. each gallon is 4 quarts, so

16.2183 gal × 4 qt/gal = 64.8732 qt

its line 2 i just added the rest or else it wouldn't let me answer