Questions: m, 2 m, 4 m, .... The first term in the sequence above is m, and each term thereafter is equal to twice the previous term. If m is an integer, which of the following could NOT be the sum of the first four terms of this sequence?

m, 2 m, 4 m, .... The first term in the sequence above is m, and each term thereafter is equal to twice the previous term. If m is an integer, which of the following could NOT be the sum of the first four terms of this sequence?

Transcript text: $\mathrm{m}, 2 \mathrm{~m}, 4 \mathrm{~m}, \ldots$. The first term in the sequence above is m , and each term thereafter is equal to twice the previous term. If $m$ is an integer, which of the following could NOT be the sum of the first four terms of this sequence?

Solution

Solution Steps

To solve this problem, we need to calculate the sum of the first four terms of the given geometric sequence. The sequence is defined as $ m, 2m, 4m, \ldots $, where each term is twice the previous term. The sum of the first four terms can be expressed as $ m + 2m + 4m + 8m $. We will calculate this sum and then determine which of the given options could not be the result of this sum.

Step 1: Define the Sequence

The sequence is defined as $ m, 2m, 4m, 8m $. Each term is twice the previous term, and we need to find the sum of the first four terms.

Step 2: Calculate the Sum

The sum of the first four terms can be expressed mathematically as: \[ S = m + 2m + 4m + 8m = 15m \]

Step 3: Determine Possible Values

Since $ m $ is an integer, the possible sums for $ m = 1, 2, \ldots, 9 $ yield the following results:

For $ m = 1 $: $ S = 15 $
For $ m = 2 $: $ S = 30 $
For $ m = 3 $: $ S = 45 $
For $ m = 4 $: $ S = 60 $
For $ m = 5 $: $ S = 75 $
For $ m = 6 $: $ S = 90 $
For $ m = 7 $: $ S = 105 $
For $ m = 8 $: $ S = 120 $
For $ m = 9 $: $ S = 135 $

Thus, the possible sums are $ 15, 30, 45, 60, 75, 90, 105, 120, 135 $.

Step 4: Identify Impossible Sums

To determine which of the given options could NOT be the sum of the first four terms, we compare the options against the calculated possible sums. Any option not in the list $ \{15, 30, 45, 60, 75, 90, 105, 120, 135\} $ is the answer.