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24 Dec 2021, 10:44
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
55%
(hard)
Question Stats:
67% (02:15) correct
33%
(02:04)
wrong
based on 290
sessions
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A sequence L is given as \(l_1, \ l_2, \ l_3, \ ⋯, \ l_n\), where n is the total number of terms. If \(l_x= l_{(x-1)}- 5\), where x is an integer such that \(2≤x ≤n\), then how many terms are there in Sequence L?
(1) \(l_n = -17\)
(2) Only 6 terms of L are greater than 9.
(1) \(l_n = -17\)
(2) Only 6 terms of L are greater than 9.
24 Dec 2021, 17:48
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ktzsikka
Breaking Down the Info:
When dealing with arithmetic sequences, there are three important elements that you may want to know:
(1) The step size of the sequence.
(2) The length of the sequence.
(3) An index in the sequence. (E.g. The 10th element is 17).
Knowing all three above would allow you to write the full sequence out.
We are given the step size is -5, we will probably need (2) and (3) in the list above.
Statement 1 Alone:
We would like to know another index and we would be able to figure out the length of the sequence. This statement is insufficient alone, however.
Statement 2 Alone:
This tells us the first 6 terms are greater than 9. Our 6th term could be 14, 13, 12, 11, or 10, however, so this is still insufficient.
Both Statements Combined:
We can backtrack the terms to find the 6th term must be -17 + 30 = 13. We have an index now so combined it is sufficient.
Answer: C
31 Dec 2021, 10:34
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Here is the OE
Solution:
Step 1: Analyse Question Stem
• L is a sequence given as \(l_1 ,l_2,l_3⋯ l_n \)
• l_x= \(l_{(x-1)}- 5\) where x is an integer and 2≤x ≤n
• Or, \(l_x- l_{(x-1)}=- 5\)
• The difference between 2 terms is constant and equal to \(-5\).Thus the sequence L is in A.P.
• Therefore, \(l_n= l_1+(n-1)*(- 5 )\)
We need to find the value of n.
Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE
Statement 1: The last term of the sequence L is -17
• \(-17 = l_1+(n-1)*(- 5)\) ………Eq.(i)
• However, both\( l_1\) and n are unknown in the above equation. So , we can’t find n from statement 1 alone.
Hence, statement 1 is not sufficient and we can eliminate answer options A and D.
Statement 2: Only 6 terms of L are greater than 9.
• Number of terms greater than 9 = 6
• But we do not know the number of terms which are less than 9.
Hence, statement 2 is also not sufficient and we can eliminate answer B.
Step 3: Analyse Statements by combining.
• From statement 1 :\( -17 = l_1+(n-1)*(- 5)\) ………Eq.(i)
• From statement 2: Number of terms greater than \(9 = 6\)
• Let us assume that -17 is \(xth\) terms from 9.
• To get the value of n, we need to find the value of x.
• To do so, we will substitute \(l_1=9\) in Eq.(i) to see how many terms are there between \(9\) and \(-17\), including \(-17\).
• However, we don’t know whether 9 is a term of sequence L or not. So, there are two possibilities.
• Possibility 1 : If 9 is not a term in L; in that case we will get a non-integral value of x and we will have to take greatest integer less than or equal to x to get the number of terms less than or equal to 9.
• Possibility 2 : If 9 is a term in L; in that case we will get an integral value of x and that will be the number of terms less than or equal to 9.
• Now, substituting l_1=9l1=9 in Eq.(i), we get,
• \(-17=9+(x-1)*(-5)\)
• Or, \(-17=14-5x\)
• Or,\( x=6.2\) which is a non-integral value
• So, the number of terms less than or equal to\( 9 = [x] = [6.2] = 6\)
Hence, the total number in the sequence \(L =6+6 =12\)
Thus, the correct answer is Option C.
Solution:
Step 1: Analyse Question Stem
• L is a sequence given as \(l_1 ,l_2,l_3⋯ l_n \)
• l_x= \(l_{(x-1)}- 5\) where x is an integer and 2≤x ≤n
• Or, \(l_x- l_{(x-1)}=- 5\)
• The difference between 2 terms is constant and equal to \(-5\).Thus the sequence L is in A.P.
• Therefore, \(l_n= l_1+(n-1)*(- 5 )\)
We need to find the value of n.
Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE
Statement 1: The last term of the sequence L is -17
• \(-17 = l_1+(n-1)*(- 5)\) ………Eq.(i)
• However, both\( l_1\) and n are unknown in the above equation. So , we can’t find n from statement 1 alone.
Hence, statement 1 is not sufficient and we can eliminate answer options A and D.
Statement 2: Only 6 terms of L are greater than 9.
• Number of terms greater than 9 = 6
• But we do not know the number of terms which are less than 9.
Hence, statement 2 is also not sufficient and we can eliminate answer B.
Step 3: Analyse Statements by combining.
• From statement 1 :\( -17 = l_1+(n-1)*(- 5)\) ………Eq.(i)
• From statement 2: Number of terms greater than \(9 = 6\)
• Let us assume that -17 is \(xth\) terms from 9.
• To get the value of n, we need to find the value of x.
• To do so, we will substitute \(l_1=9\) in Eq.(i) to see how many terms are there between \(9\) and \(-17\), including \(-17\).
• However, we don’t know whether 9 is a term of sequence L or not. So, there are two possibilities.
• Possibility 1 : If 9 is not a term in L; in that case we will get a non-integral value of x and we will have to take greatest integer less than or equal to x to get the number of terms less than or equal to 9.
• Possibility 2 : If 9 is a term in L; in that case we will get an integral value of x and that will be the number of terms less than or equal to 9.
• Now, substituting l_1=9l1=9 in Eq.(i), we get,
• \(-17=9+(x-1)*(-5)\)
• Or, \(-17=14-5x\)
• Or,\( x=6.2\) which is a non-integral value
• So, the number of terms less than or equal to\( 9 = [x] = [6.2] = 6\)
Hence, the total number in the sequence \(L =6+6 =12\)
Thus, the correct answer is Option C.
