Problem Solving

In the sequence 1, 2, 4, 8, 16, 32, ..., each term after the first is twice the previous term.
What is the sum of the 16th, 17th, and 18th terms in the sequence?

A. 218 ...its 2 exponent 18
B. 3(217)...its 3* (2 exponent 17)
C. 7(216) ...its 7*( 2 exponent 16)
D. 3(216) ...its 3* ( 2exponent 16)
E. 7(215)...its 7* ( 2 exponent 15)


Answer E

phelps wrote:In the sequence 1, 2, 4, 8, 16, 32, ..., each term after the first is twice the previous term.
What is the sum of the 16th, 17th, and 18th terms in the sequence?

A. 218 ...its 2 exponent 18
B. 3(217)...its 3* (2 exponent 17)
C. 7(216) ...its 7*( 2 exponent 16)
D. 3(216) ...its 3* ( 2exponent 16)
E. 7(215)...its 7* ( 2 exponent 15)


Answer E

The series shown above is geometric progression..

Geometric Progression: when the ratio of any two consecutive terms remains constant...for ex as u can see here in this question...

2/1 = 4/2 = 8/4.....= 2 (constant)

In GP, formula to calculate nth term is

nth term = ar^(n-1)

a = first term of the sequence
r = ratio b/w any two consecutive terms
n = the term u have to find out

so by applying the formula

16th term = 1*2^(16-1) => 2^15

similarly, 17th term = 2^16

and, 18th term = 2^17

if u add these three terms u get Op E as an answer

Hope this help!!

phelps wrote:In the sequence 1, 2, 4, 8, 16, 32, ..., each term after the first is twice the previous term.
What is the sum of the 16th, 17th, and 18th terms in the sequence?

A. 218 ...its 2 exponent 18
B. 3(217)...its 3* (2 exponent 17)
C. 7(216) ...its 7*( 2 exponent 16)
D. 3(216) ...its 3* ( 2exponent 16)
E. 7(215)...its 7* ( 2 exponent 15)

Hi,

since the first term is 1, and since we double to find the next term, we can quickly see that the nth term of the sequence is simply 2^(n-1). In other words:

term1 = 2^0
term2 = 2^1
term3 = 2^2

and so on.

So, the 16th, 17th and 18th terms will be:

2^15, 2^16 and 2^17.

Next, we need to take the sum of these terms. We have a simple rule for multiplying exponents with the same base:

x^a * x^b = x^(a+b)

for dividing exponents with the same base:

x^a/x^b = x^a-b)

and for raising a power to another power:

(x^a)^b = x^(a*b)

but there's no simple rule for adding or subtracting exponents with the same base. The only way we can add or subtract is if both the base and the exponent are equal - and we do so simply by adding or subtracting the coefficients. For example:

3(x^4) + 2(x^4) = 5(x^4).

So, if we want to add our three terms, we need to equalize the exponents. To do so, we need to factor down the bigger exponents to match the smallest one.

Since 2^15 is the smallest, we leave that alone.

2^16 = 2^1 * 2^15 = 2(2^15)

2^17 = 2^2 * 2^15 = 4(2^15)

Now we can sum our terms:

2^15 + 2^16 + 2^17 = 1(2^15) + 2(2^15) + 4(2^15) = 7(2^15)... choose (E).

phelps wrote:In the sequence 1, 2, 4, 8, 16, 32, ..., each term after the first is twice the previous term.
What is the sum of the 16th, 17th, and 18th terms in the sequence?

A. 218 ...its 2 exponent 18
B. 3(217)...its 3* (2 exponent 17)
C. 7(216) ...its 7*( 2 exponent 16)
D. 3(216) ...its 3* ( 2exponent 16)
E. 7(215)...its 7* ( 2 exponent 15)


Answer E

Determine the pattern.

1st term = 2^0.
2nd term = 2^1.
3rd term = 2^2.
The exponent for the nth term is always n-1.

Thus:
16th term = 2^15.
17th term = 2^16.
18th term = 2^17.

2^15 + 2^16 + 2^17
= 2^15(1 + 2 + 2^2)
= 2^15 * 7.

The correct answer is E.