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# A certain sequence starts with term_1

tagged by: Brent@GMATPrepNow

### Top Member

#### A certain sequence starts with term_1

A certain sequence starts with term_1

For any term in the sequence, term_n = 16^(2n - 1)

If the PRODUCT of the first k terms of the sequence is 2^1600, what is the value of k?

A) 5
B) 10
C) 20
D) 40
E) 80

Can some experts show me the best solution?

OA C

### GMAT/MBA Expert

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lheiannie07 wrote:
A certain sequence starts with term_1

For any term in the sequence, term_n = 16^(2n - 1)

If the PRODUCT of the first k terms of the sequence is 2^1600, what is the value of k?

A) 5
B) 10
C) 20
D) 40
E) 80
First notice that 2n - 1, will be ODD for all integer values of n. For example:
If n = 1, then 2n - 1 = 2(1) - 1 = 1
If n = 2, then 2n - 1 = 2(2) - 1 = 3
If n = 3, then 2n - 1 = 2(3) - 1 = 5
If n = 4, then 2n - 1 = 2(4) - 1 = 7
.
.
.
etc.

Now notice what happens when we add consecutive ODD numbers (starting with 1)
The first 1 ODD number: 1 = 1 (and 1 = 1²)
The first 2 ODD numbers: 1 + 3 = 4 (and 4 = 2²)
The first 3 ODD numbers: 1 + 3 + 5 = 9 (and 9 = 3²)
The first 4 ODD numbers: 1 + 3 + 5 + 7 = 16 (and 16 = 4²)
The first 5 ODD numbers: 1 + 3 + 5 + 7 + 9 = 25 (and 25 = 5²)
.
.
.
In general, the sum of the first k ODD numbers = k²

Now onto the question!!!

term_n = 16^(2n - 1)
term_1 = 16^(2(1) - 1) = 16^1
term_2 = 16^(2(2) - 1) = 16^3
term_3 = 16^(3(3) - 1) = 16^5
term_4 = 16^(2(4) - 1) = 16^7
etc

So, the PRODUCT of the first k terms = (16^1)(16^3)(16^5)(16^7)(16^9). . . (16^??)
When we multiply powers with the same base, we ADD the exponents.
So, the PRODUCT of the first k terms = 16^(1 + 3 + 5 + 7 + . . . ??)

Notice that the exponent here is equal to the SUM of the first k ODD numbers.
Well, we already know that the sum of the first k ODD numbers = k²
So, the PRODUCT of the first k terms = 16^()

We're told that the PRODUCT of the first k terms is 2^1600
So, we can write: 16^() = 2^1600

We need the same base, so let's rewrite 16 as 2^4
We get: (2^4)^() = 2^1600
Apply power of a power law: 2^(4k²) = 2^1600
This means that 4k² = 1600
Divide both sides by 4 to get: k² = 400
Solve: k = 20 or -20
Since -20 makes no sense, we know that k = 20

In other words, the PRODUCT of the first 20 terms of the sequence is 2^1600,

Cheers,
Brent

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lheiannie07 wrote:
A certain sequence starts with term_1

For any term in the sequence, term_n = 16^(2n - 1)

If the PRODUCT of the first k terms of the sequence is 2^1600, what is the value of k?

A) 5
B) 10
C) 20
D) 40
E) 80
[/i]
Since the formula for the sequence is in terms of 16, rephrase the sum in terms of 16:
2¹⁶⁰⁰ = 2⁴*⁴⁰⁰ = 16⁴⁰⁰.

Calculate the first few terms and look for a PATTERN.
n=1 --> 16²*¹¯¹ = 16¹.
n=2 --> 16²*²¯¹ = 16³.
n=3 --> 16³*²¯¹ = 16⁵.
Product of the first 3 terms = 16¹16³16⁵ = 16¹⁺³⁺⁵ = 16⁹.
Implication of the exponent in blue:
Product of the first 3 terms = 16^(sum of the first 3 positive odd integers).
By extension:
Product of the first k terms = 16^(sum of the first k positive odd integers).

To yield the desired sum -- 16⁴⁰⁰ -- the sum of the first k positive integers must be equal to 400.
Sum of the first k positive odd integers = k².
Examples:
Sum of the first 2 positive odd integers = 1+3 = 4 = 2².
Sum of the first 3 positive odd integers = 1+3+5 = 9 = 3².
Sum of the first 4 positive odd integers = 1+3+5+7 = 16 = 4².
If there are k terms, the sum = k².
Since we want the sum of the first k positive odd integers to be 400, we get:
k² = 400
k = 20.

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