The sum of the first 10 terms of a geometric sequence is 1023. The common ratio is 2. What is the fifth term of the sequence?

The fifth term of the sequence is 16

Step-by-step explanation:

Firstly, we need to write the mathematical expression for the sum of terms in a geometric sequence.

Mathematically, the sum of terms S is calculated as follows;

S = a(r^n -1)/r-1

where a is the first term of the sequence, r is the common ratio, and n is the number of terms. From the question, we can see we have 10 as the number of terms here, 1023 as the sum, and 2 as the common ratio. We thus, plug these values into the equation above.

1023= a(2^10 - 1)/2-1

1023= a(1024-1)/1

1023 = 1023a

a = 1023/1023

a = 1

In the question we are told to find the fifth term:

mathematically, the nth term of a geometric sequence can be calculated using the formula Tn = ar^(n-1). For the fifth term, n = 5 and thus T5 = ar^4

T5 = 1 * 2^4 = 1 * 16 = 16

T5 =

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