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An arithmetic sequence is such that the division of the fifth term by

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An arithmetic sequence is such that the division of the fifth term by [#permalink] New post   28 Dec 2023, 10:07
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Data Insights (DI) Butler 2023-24 [Question #173, Date: Dec-27-2023] [Click here for Details]


An arithmetic sequence is such that the division of the fifth term by the third term gives a quotient of 1 and leaves a remainder of 10, while the product of the first and third terms equals 24.

In the table, select a value for the first term, \(a_{1}\) of the sequence, and a value for the difference, \(d\), between successive terms of the sequence, that together are consistent with the given information. Make only one selection in each column.
\(a_{1}\) \(d\)
2
3
5
7
12
22
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Re: An arithmetic sequence is such that the division of the fifth term by [#permalink] New post   28 Dec 2023, 11:03
Let terms be a1,a2... a5
Given a5=a3×1+10
a1×a3=24
a1 =2
a2=7
a3=12
a4=17
a5=22
difference is 5
first term is 2

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Re: An arithmetic sequence is such that the division of the fifth term by [#permalink] New post   03 Jan 2024, 04:20
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An arithmetic sequence is such that the division of the fifth term by the third term gives a quotient of 1 and leaves a remainder of 10, while the product of the first and third terms equals 24.

Now, \(a_5=a_3*1+10=a_3+10\)…..(1)
But we also know \(a_5=a_4+d=a_3+d+d=a_3+2d\)……(2)

From 1 and 2 we get \(2d=10…..d=5\)

Also, \(a_1*a_3=24……..a_1(a_1+10)=24=2*12…….a_1=2\)

Solution
\(a_1=2\)
\(d=5\)
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Re: An arithmetic sequence is such that the division of the fifth term by [#permalink] New post   10 Apr 2024, 23:54
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\(­a_1\)
\(a_2\) = \(a_1\) + d
\(a_3\) = \(a_1\) + 2d
\(a_4\) = \(a_1\) + 3d
\(a_5\) = \(a_1\) + 4d
  • \(a_5\) = \(a_3\) + 2d
  • The division of the fifth term by the third term gives a quotient of 1 and leaves a remainder of 10 => \(a_5\) = \(a_3\) + 10
=> 2d = 10
=> d = 5


24 = \(­a_1\) * \(a_3\)

24 = 1 * 24 = 2 * 12  = 3 * 8

=> \(­a_1\) = 2; \(a_3\) = 12 (diffference is 10 = 2d)

\(­a_1\) = 2
d = 5



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Re: An arithmetic sequence is such that the division of the fifth term by [#permalink]
10 Apr 2024, 23:54
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