How many terms of the sequence are greater than 10?
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Statement 1 is clearly insufficient.a(1), a(2),...., a(15)
In the sequence shown, a(n) = a(n-1) + k, where 2≤n≤15 and k is a nonzero constant. How many of the terms in the sequence are greater than 10?
1) a(1)=24
2) a(8)=10
Statement 2: a₈ = 10.
If k>0, then the sequence is INCREASING: each term in the sequence is GREATER than the preceding term.
In this case, a₉...a�₅ -- a total of 7 terms -- will be greater than 10.
If k<0, then the sequence is DECREASING: each term in the sequence is LESS than the preceding term.
In this case, a�...a₇ -- a total of 7 terms -- will be greater than 10.
In each case, the number of terms greater than 10 = 7.
SUFFICIENT.
The correct answer is B.
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Target question: How many term in the sequence a1,a2,a3...a15 are greater than 10?a1,a2,a3...a15
In the sequence shown, a(n) = a(n-1)+k, where 2 < n < 15, and k is a nonzero constant. How many of the terms in the sequence are greater than 10?
(1) a1 = 24
(2) a8 = 10
Given: a(n) = a(n-1)+k, where 2 < n < 15
In other words, each term is derived by taking the term before it and adding k
IMPORTANT: Keep in mind that k can be either a positive or negative number. So, the sequence may be increasing (e.g., 5, 7, 9, 11...) or it may be decreasing (e.g., 20, 15, 10, ...)
Statement 1: a1 = 24
The 1st term is 24, but since we don't know the value of k, there's no way to determine the terms in the sequence that are greater than 10
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: a8 = 10
Let's consider the 2 possible cases for k (k is POSITIVE or k is NEGATIVE)
case a: k is POSITIVE
This means that the sequence is INCREASING.
In other words, term1 < term2 < term3, etc.
The 8th term is 10, which means every term after the 8th term must be greater than 10.
So, terms 9, 10, 11, 12, 13, 14 and 15 are greater than 10.
This means that 7 terms in the sequence are greater than 10
case b: k is NEGATIVE.
This means that the sequence is DECREASING.
In other words, term1 > term2 > term3, etc.
The 8th term is 10, which means every term before the 8th term must be greater than 10.
So, terms 1, 2, 3, 4, 5, 6 and 7 are greater than 10.
This means that 7 terms in the sequence are greater than 10
Since BOTH cases yield the SAME answer to the target question, we can be certain that 7 terms in the sequence are greater than 10
Statement 2 is SUFFICIENT
Answer = B
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Brent
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You could think of the sequence as
z, z+k, z+2k, z+3k, ..., z+14k
S2 tells us that a8 = 10 = z + 7k, so if k is negative, z + 7k > z + 8k > z + 9k > ... > z + 14k, and all the terms from a9 to a15 are less than a8, i.e. less than 10.
By the same token, if k is positive, then z + 7k < z + 8k < z + 8k < z + 9k < ... < z + 14k, and all the terms from a9 to a15 are GREATER than a8, i.e. greater than 10.
In either case, we have 7 terms less than 10, one term equal to 10, and 7 terms greater than 10.
z, z+k, z+2k, z+3k, ..., z+14k
S2 tells us that a8 = 10 = z + 7k, so if k is negative, z + 7k > z + 8k > z + 9k > ... > z + 14k, and all the terms from a9 to a15 are less than a8, i.e. less than 10.
By the same token, if k is positive, then z + 7k < z + 8k < z + 8k < z + 9k < ... < z + 14k, and all the terms from a9 to a15 are GREATER than a8, i.e. greater than 10.
In either case, we have 7 terms less than 10, one term equal to 10, and 7 terms greater than 10.