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
A certain sequence starts with term_1
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Since the formula for the sequence is in terms of 16, rephrase the sum in terms of 16: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
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2¹��� = 2�*��� = 16���.
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 = 16�.
Product of the first 3 terms = 16¹16³16� = 16¹�³�� = 16�.
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 -- 16��� -- 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.
The correct answer is C.
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First notice that 2n - 1, will be ODD for all integer values of n. For example: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
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^(k²)
We're told that the PRODUCT of the first k terms is 2^1600
So, we can write: 16^(k²) = 2^1600
We need the same base, so let's rewrite 16 as 2^4
We get: (2^4)^(k²) = 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,
Answer: C
Cheers,
Brent