How many solutions does this equation have? a + 3 + 2a = -1 + 3a + 4

There are infinitely many solutions of this equation.

Step-by-step explanation:

The given equation :

First we try to solve the above equation  by combining the like terms (by reordering them), we get

Subtracting 3a from both sides (by using subtracting property of equality), we get

, which is absolutely a correct fact.

That means, the equation has infinitely many solutions as a can take any real value .

Thus , there are infinitely many solutions of this equation.

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