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# If the a-th term of a certain sequence is given by (-1)^a + (-0.1)^a

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Math Expert
Joined: 02 Sep 2009
Posts: 64242
If the a-th term of a certain sequence is given by (-1)^a + (-0.1)^a  [#permalink]

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29 Jul 2019, 04:12
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Difficulty:

65% (hard)

Question Stats:

55% (02:23) correct 45% (02:19) wrong based on 71 sessions

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If the a-th term of a certain sequence is given by $$(-1)^a + (-0.1)^a,$$ then the sum of the terms from 1 to 5 is

A. -1.11111
B. -1.09091
C. 0
D. 0.90909
E. 1.11111

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Re: If the a-th term of a certain sequence is given by (-1)^a + (-0.1)^a  [#permalink]

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29 Jul 2019, 08:35
Bunuel wrote:
If the a-th term of a certain sequence is given by $$(-1)^a + (-0.1)^a,$$ then the sum of the terms from 1 to 5 is

A. -1.11111
B. -1.09091
C. 0
D. 0.90909
E. 1.11111

a=1 gives us -1-0.1 which is -1.1
a=2 gives us 1+0.1 which is 1.01
a=3 gives us -1-0.001 which is -1.001
a=4 gives us 1+0.0001 which is 1.0001
a=5 gives us -1-0.00001 which is -1.00001

Total is -1.09091.

What is the shortcut here Please?
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Re: If the a-th term of a certain sequence is given by (-1)^a + (-0.1)^a  [#permalink]

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29 Jul 2019, 08:46
TheNightKing wrote:
Bunuel wrote:
If the a-th term of a certain sequence is given by $$(-1)^a + (-0.1)^a,$$ then the sum of the terms from 1 to 5 is

A. -1.11111
B. -1.09091
C. 0
D. 0.90909
E. 1.11111

a=1 gives us -1-0.1 which is -1.1
a=2 gives us 1+0.1 which is 1.01
a=3 gives us -1-0.001 which is -1.001
a=4 gives us 1+0.0001 which is 1.0001
a=5 gives us -1-0.00001 which is -1.00001

Total is -1.09091.

What is the shortcut here Please?

Posted from my mobile device

I think it is easier to work with huristics.

Clearly odd power implies the first for (-1) ^a terms cancel out.
The other terms are decreasing negatives with each increase in power. The first will be -0.1 and the sum of the rest will a positive decimal less than -0.1

Only answer matching such a solution is B
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Re: If the a-th term of a certain sequence is given by (-1)^a + (-0.1)^a  [#permalink]

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06 Aug 2019, 16:54
Bunuel wrote:
If the a-th term of a certain sequence is given by $$(-1)^a + (-0.1)^a,$$ then the sum of the terms from 1 to 5 is

A. -1.11111
B. -1.09091
C. 0
D. 0.90909
E. 1.11111

When a is 1:

-1 + (-0.1)

When a is 2:

1 + 0.01

When a is 3:

-1 + (-0.001)

When a is 4:

1 + (0.0001)

When a is 5:

-1 + (-0.00001)

We see that the sum of integer parts of the first 5 terms is:

-1 + 1 + (-1) + 1 + (-1) = -1

And the sum of the decimal parts of the first 5 terms is:

-0.1 + 0.01 + (-0.001) + 0.0001 + (-0.00001) = -0.09 + (-0.0009) + (-0.00001) = -0.09091

Therefore, the sum of the first 5 terms is:

-1 + (-0.09091) = -1.09091

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Re: If the a-th term of a certain sequence is given by (-1)^a + (-0.1)^a   [#permalink] 06 Aug 2019, 16:54