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4 years ago

How do you solve the first problem? (About the geometric series, and finding the first term and common ratio)

Mathematics

1 answer:

kogti [31]4 years ago

4 0

Answer:

First term = 5.

Common ratio = -3.

Step-by-step explanation:

Third term = a1 r^2 = 45. As r is an integer 3 or -3 is a possible value of r.

If r = 3 then the first term a1 = 45 / 9 = 5.

Substitute a1 = 5 and r = 3 into the formula for the sum of 7 terms which is given as 2735.

Try r = 3:

S7 = 5 * (3^7 - 1) / (3 - 1) = 5465 so r can't be equal to 3.

Try r = -3:-

If r = -3 then a1 = 5 so

S7 = 5 * ((-3)^7 - 1) / (-3-1)

= 5 * 547

= 2735 which is the value given.

So r = -3 and a1 = 5.

You might be interested in

Evaluate each finite series for the specified number of terms. 1+2+4+...;n=5

zaharov [31]

Answer:

Step-by-step explanation:

The series are given as geometric series because these terms have common ratio and not common difference.

Our common ratio is 2 because:

1*2 = 2

2*2 = 4

The summation formula for geometric series (r ≠ 1) is:

$\displaystyle \large{S_n=\frac{a_1(r^n-1)}{r-1}}$ or $\displaystyle \large{S_n=\frac{a_1(1-r^n)}{1-r}}$

You may use either one of these formulas but I’ll use the first formula.

We are also given that n = 5, meaning we are adding up 5 terms in the series, substitute n = 5 in along with r = 2 and first term = 1.

$\displaystyle \large{S_5=\frac{1(2^5-1)}{2-1}}\\\displaystyle \large{S_5=\frac{2^5-1}{1}}\\\displaystyle \large{S_5=2^5-1}\\\displaystyle \large{S_5=32-1}\\\displaystyle \large{S_5=31}$

Therefore, the solution is 31.

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Summary

If the sequence has common ratio then the sequence or series is classified as geometric sequence/series.

Common Ratio can be found by either multiplying terms with common ratio to get the exact next sequence which can be expressed as $\displaystyle \large{a_{n-1} \cdot r = a_n}$ meaning “previous term times ratio = next term” or you can also get the next term to divide with previous term which can be expressed as:

$\displaystyle \large{r=\frac{a_{n+1}}{a_n}}$

Once knowing which sequence or series is it, apply an appropriate formula for the series. For geometric series, apply the following three formulas:

$\displaystyle \large{S_n=\frac{a_1(r^n-1)}{r-1}}\\\displaystyle \large{S_n=\frac{a_1(1-r^n)}{1-r}}$

Above should be applied for series that have common ratio not equal to 1.

$\displaystyle \large{S_n=a_1 \cdot n}$

Above should be applied for series that have common ratio exactly equal to 1.

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Topics

Sequence & Series — Geometric Series

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Others

Let me know if you have any doubts about my answer, explanation or this question through comment!

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7 0

3 years ago

The number 46 comma 462 comma 555 46,462,555 rounded to the nearest million is

Nataliya [291]

The nearest million is 46 million since if the number to the right of "nearest so and so" is less than 5, you round down but if it is 5 or more then you round up.
Hope this helps!

8 0

4 years ago

Two trains leave the station at the same, one heading west and the other east. The westbound train travels at 55 miles per hour.

Mars2501 [29]

The time that taken for the two trains is 1.2 hours.

The computation of the time taken for the two trains is shown below:

Given that

The westbound train travels 55 miles per hour.

And, the eastbound train travels 75 miles per hour.

So, the total speed is

= 55 miles per hour + 75 miles per hour

= 130 miles per hour