Identify if each sequence is arithmetic, geometric, or neither. 7 , 10 , 16 , 25 , 37 , … 7,10,16,25,37,\ldots

this problem, we need to identify if this sequence, seven, ten, sixteen, twenty five, 37 is arithmetic, geometric, or neither. Now remember, an arithmetic sequence, we are adding or subtracting to go from one number to the next, and it's always the same amount by our common difference. And for a geometric sequence, we are multiplying to go from one number to the next by our common ratio. So let's check if this is arithmetic first. We can do that by taking one term and subtracting it from the previous term, seven minus 10 is three, and do that again for all of our terms. 16 minus 10 is six, and already we can see that since these two numbers are not the same, this is not going to be arithmetic. So next, let's check for geometric. We can do the same thing, but instead of subtracting, we're going to divide. So 10 over seven is equal to about 1.4. 16 over 10 is equal to 1.6, and again, since these two are not the same, this means this is not going to be geometric either, which means that for this problem, our correct choice is neither.

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1. Review of Real Numbers2h 39m

Simplifying Fractions
30m

Evaluating Exponents
29m

Evaluating Expressions
15m

Translating Phrases to Expressions
16m

Translating Sentences to Equations
9m

The Distributive Property
17m

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40m

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41m

The Multiplication and Division Properties of Equality
30m

Solving Linear Equations
1h 14m

Linear Inequalities in One Variable
1h 11m

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37m

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21m

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59m

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43m

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44m

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24m

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23m

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38m

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22m

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41m

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15m

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45m

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10m

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18m

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18m

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28m

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6m

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21m

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20m

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33m

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34m

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38m

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11m

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37m

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33m

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41m

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39m

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25m

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19m

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32m

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32m

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44m

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42m

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28m

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23m

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27m

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47m

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47m

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19m

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53m

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18m

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40m

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40m

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27m

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26m

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11m

15. Sequences, Series, and the Binomial Theorem

Geometric Sequences

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15. Sequences, Series, and the Binomial Theorem

Geometric Sequences

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Identify if each sequence is arithmetic, geometric, or neither.
$7 comma 10 comma 16 comma 25 comma 37 comma dot dot dot$

Arithmetic

Geometric

Neither

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Problema precedenteProssimo problema

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First, recall the definitions: an arithmetic sequence has a constant difference between consecutive terms, and a geometric sequence has a constant ratio between consecutive terms.

Calculate the differences between consecutive terms: $10 - 7 = 3$, $16 - 10 = 6$, $25 - 16 = 9$, and $37 - 25 = 12$. Since these differences are not the same, the sequence is not arithmetic.

Next, calculate the ratios between consecutive terms: $\frac{10}{7}$, $\frac{16}{10}$, $\frac{25}{16}$, and $\frac{37}{25}$. Since these ratios are not constant, the sequence is not geometric.

Since the sequence is neither arithmetic nor geometric, it falls into the category of neither.

Therefore, the sequence does not have a constant difference or ratio, confirming it is neither arithmetic nor geometric.