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Zarrin [17]
3 years ago
5

Please help me with this

Mathematics
1 answer:
MA_775_DIABLO [31]3 years ago
3 0

Answer:

13.7 cm²

Step-by-step explanation:

area of yellow region = area of square - area of 4 quarter circles

area of square = 8² = 64

area of 4 congruent quarter circles = area of circle

area of circle with radius = 4

A = π × 4² = 16π

yellow region = 64 - 16π ≈ 13.7 cm²

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Answer:

The first three terms of the geometry sequence would be 1, 5, and 25.

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<h3>1.</h3>

Let d denote the common difference of the arithmetic sequence.

Let a_1 denote the first term of the arithmetic sequence. The expression for the nth term of this sequence (where n\! is a positive whole number) would be (a_1 + (n - 1)\, d).

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  • The third term of this arithmetic sequence would be a_1 + (3 - 1)\, d = 1 + 2\, d.
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The common ratio of a geometric sequence is ratio between consecutive terms of that sequence. Let r denote the ratio of the geometric sequence in this question.

Ratio between the second term and the first term of the geometric sequence:

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Ratio between the third term and the second term of the geometric sequence:

\displaystyle r = \frac{1 + 12\, d}{1 + 2\, d}.

Both (1 + 2\, d) and \left(\displaystyle \frac{1 + 12\, d}{1 + 2\, d}\right) are expressions for r, the common ratio of this geometric sequence. Hence, equate these two expressions and solve for d, the common difference of this arithmetic sequence.

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Hence, the first term, the third term, and the thirteenth term of the arithmetic sequence would be 1, (1 + (3 - 1) \times 2) = 5, and (1 + (13 - 1) \times 2) = 25, respectively.

These three terms (1, 5, and 25, respectively) would correspond to the first three terms of the geometric sequence. Hence, the common ratio of this geometric sequence would be r = 25 /5 = 5.

<h3>2.</h3>

Let a_1 and r denote the first term and the common ratio of a geometric sequence. The sum of the first n terms would be:

\displaystyle \frac{a_1 \, \left(1 - r^{n}\right)}{1 - r}.

For the geometric sequence in this question, a_1 = 1 and r = 25 / 5 = 5.

Hence, the sum of the first n = 7 terms of this geometric sequence would be:

\begin{aligned} & \frac{a_1 \, \left(1 - r^{n}\right)}{1 - r}\\ &= \frac{1 \times \left(1 - 2^{7}\right)}{1 - 2} \\ &= \frac{(1 - 128)}{(-1)} = 127 \end{aligned}.

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