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15 Jul 2022, 08:00
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E
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
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The infinite sequence of integers \(a_1\), \(a_2\), …, \(a_n\), … is such that \(a_1 = -17\) and \(a_{n-1} - 7 < a_n < a_{n-1} - 2\) for \(n > 1\). If \(a_x=-53\), then how many different values can x take?
A. 1
B. 4
C. 5
D. 6
E. 7
A. 1
B. 4
C. 5
D. 6
E. 7
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Updated on: 16 Jul 2022, 08:22
Originally posted by ThatDudeKnows on 15 Jul 2022, 08:33.
Last edited by ThatDudeKnows on 16 Jul 2022, 08:22, edited 1 time in total.
Last edited by ThatDudeKnows on 16 Jul 2022, 08:22, edited 1 time in total.
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Bunuel
\(a_1 = -17\)
\(a_2\):
\(a_1-7 < a_2 <a_1-2\)
\(-17-7 < a_2 <-17-2\)
\(-24 < a_2 <-19\)
\(a_2\): {-23, -22, -21, -20}
\(a_3\):
\(-23-7 < a_3 <-20-2\)
\(-30 < a_3 <-22\)
\(a_3\): {-29, -28, -27, -26, -25, -24, -23}
Bottom end of the range moves to the left by 6 and top end of the range moves to the left by three each iteration.
\(a_4\): {-35,...-26}
\(a_5\): {-41,...-29}
\(a_6\): {-47,...-32}
\(a_7\): {-53,...-35}
\(a_8\): {-56,...-38}
\(a_9\): {,...-41}
\(a_9\): {,...-44}
\(a_10\): {,...-47}
\(a_11\): {,...-50}
\(a_12\): {,...-53}
That's seven.
Answer choice E.
15 Jul 2022, 18:21
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Answer: E
Infinite sequence of integers is a1, a2, a3, .... ,an
a1 = -17
a(n-1) - 7 < an < a(n-1) - 2 for n > 1
=> a1 - 7 < a2 < a1 - 2
Similarly, a2 - 7 < a3 < a2 - 2
a3 - 7 < a4 < a3 - 2
and so on...
a1 = -17, we get -24 < a2 < -19
i.e. a2 can have 4 values -23, -22, -21, -20.
To get the maximum range for a3, for LHS consider a2 = -23 and for RHS consider a2 = -20
=> -23-7 < a3 < -20 - 2
=> -30 < a3 < -22
i.e. a3 can have integer values -29, -28, -27,...., -24, -23
Similarly, -36 < a4 < -25
i.e. a4 can have integer values -35, -34, -33,....,-27, -26
For every next an, the range will increase by -6 on left side and increase by -3 on right side.
=> -36 < a4 < -25
=> -42 < a5 < -28
=> -48 < a6< -31
=> -54 < a7 < -34
=> -60 < a8 < -37
=> -66 < a9 < -40
=> -72 < a10 < -43
=> -78 < a11 < -46
=> -84 < a12 < -49
=> -90 < a13 < -52
=> -96 < a14 < -55
=> -102 < a15 < -58
Now for ax = −53, we can see from above that -53 will be in range for a8, a9, ... , a13 and a14
so x can take values 8, 9, 10, 11, 12, 13 and 14.
Total 7 values.
Infinite sequence of integers is a1, a2, a3, .... ,an
a1 = -17
a(n-1) - 7 < an < a(n-1) - 2 for n > 1
=> a1 - 7 < a2 < a1 - 2
Similarly, a2 - 7 < a3 < a2 - 2
a3 - 7 < a4 < a3 - 2
and so on...
a1 = -17, we get -24 < a2 < -19
i.e. a2 can have 4 values -23, -22, -21, -20.
To get the maximum range for a3, for LHS consider a2 = -23 and for RHS consider a2 = -20
=> -23-7 < a3 < -20 - 2
=> -30 < a3 < -22
i.e. a3 can have integer values -29, -28, -27,...., -24, -23
Similarly, -36 < a4 < -25
i.e. a4 can have integer values -35, -34, -33,....,-27, -26
For every next an, the range will increase by -6 on left side and increase by -3 on right side.
=> -36 < a4 < -25
=> -42 < a5 < -28
=> -48 < a6< -31
=> -54 < a7 < -34
=> -60 < a8 < -37
=> -66 < a9 < -40
=> -72 < a10 < -43
=> -78 < a11 < -46
=> -84 < a12 < -49
=> -90 < a13 < -52
=> -96 < a14 < -55
=> -102 < a15 < -58
Now for ax = −53, we can see from above that -53 will be in range for a8, a9, ... , a13 and a14
so x can take values 8, 9, 10, 11, 12, 13 and 14.
Total 7 values.
