(a) Compute the (approximate) values of the terms in thesequence1.01101, 1.0011001

Chapter 1, Problem 67

(choose chapter or problem)

(a) Compute the (approximate) values of the terms in the sequence

$1.01^{101}, 1.001^{1001}, 1.0001^{10001}, 1.00001^{100001}$

$1.000001^{1000001}, 1.0000001^{10000001} \ldots$

What number do these terms appear to be approaching?

(b) Use Equation (7) to verify your answer in part (a).

(c) Let $1 \leq a \leq 9$ denote a positive integer. What number is approached more and more closely by the terms in the following sequence?

$1.01^{a 0 a}, 1.001^{a 00 a}, 1.0001^{a 000 a}, 1.00001^{a 0000 a}$

$1.000001^{a 00000 a}, 1.0000001^{a 000000 a} \ldots$

(The powers are positive integers that begin and end with the digit a and have 0 's in the remaining positions).

Equation Transcription:

${1.01}^{101},{1.001}^{1001},{1.0001}^{10001},{1.00001}^{100001}\$

${1.000001}^{1000001},{1.0000001}^{10000001}\ldots{}$

$1\ \leq{}\ a\ \leq{}\ 9$

${1.01}^{a0a},{1.001}^{a00a},{1.0001}^{a000a},{1.00001}^{a0000a}$

${1.000001}^{a00000a},{1.0000001}^{a000000a}\ldots{}$

Text Transcription:

1.01^101, 1.001^1001, 1.0001^10001, 1.00001^100001

1.000001^1000001,1.0000001^10000001...

1 less than or equal to a less than or equal to 9

1.01^a0a,1.001^a00a,1.0001^a000a,1.00001^a0000a

1.000001^a00000a,1.0000001^a000000a...

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