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In the sequence of non-zero numbers t1, t2, t3, . . . tn, . . . , the
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11 Jan 2024, 16:29
11 Jan 2024, 16:29
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In the sequence of non-zero numbers \(t_1, t_2, t_3, . . . t_n, . . .\) , the value of \(t_{(n+1)}=\frac{t_n}{2}\) for all positive integers n. If \(t_1-t_5=\frac{15}{16}\), what is the value of \(t_5\)?
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\(\frac{1}{16}\)
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In the sequence of non-zero numbers t1, t2, t3, . . . tn, . . . , the
[#permalink]
12 Jan 2024, 12:07
12 Jan 2024, 12:07
1
Find \(t_5\) in terms of \(t_1\):
\(t_2 = \frac{t_1}{2}\)
\(t_3 = \frac{t_2}{2} = \frac{t_1}{2} / 2 = \frac{t_1}{4}\)
\(t_4 = \frac{t_3}{2} = \frac{t_1}{8}\)
\(t_5 = \frac{t_4}{2} = \frac{t_1}{16}\)
\(t_1 - t_5 = \frac{15}{16}\)
Substitute in \(t_5 = \frac{t_1}{16}\):
\(t_1 - \frac{t_1}{16} = \frac{15}{16}\)
\(t_1 (\frac{16}{16}- \frac{1}{16}) = \frac{15}{16}\)
\(t_1 (\frac{15}{16}) = \frac{15}{16}\)
\(t_1 = 1\)
\(t_5 = \frac{1}{16}\)
\(t_2 = \frac{t_1}{2}\)
\(t_3 = \frac{t_2}{2} = \frac{t_1}{2} / 2 = \frac{t_1}{4}\)
\(t_4 = \frac{t_3}{2} = \frac{t_1}{8}\)
\(t_5 = \frac{t_4}{2} = \frac{t_1}{16}\)
\(t_1 - t_5 = \frac{15}{16}\)
Substitute in \(t_5 = \frac{t_1}{16}\):
\(t_1 - \frac{t_1}{16} = \frac{15}{16}\)
\(t_1 (\frac{16}{16}- \frac{1}{16}) = \frac{15}{16}\)
\(t_1 (\frac{15}{16}) = \frac{15}{16}\)
\(t_1 = 1\)
\(t_5 = \frac{1}{16}\)
gmatclubot
In the sequence of non-zero numbers t1, t2, t3, . . . tn, . . . , the [#permalink]
12 Jan 2024, 12:07
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