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in a geometric sequence, the fourth term is 8 times the first term. the sum of the first 10 terms is 2557.5. find the 10th term of this sequence.

Respuesta :

Answer: 1280

Step-by-step explanation:

The fourth term is 8 times the first term.

The fourth term = [tex]ar^3[/tex]

[tex]ar^{3} = 8a[/tex]

To find the common ratio:

∴ [tex]\frac{ar^3}{a} = \frac{8a}{a}\\[/tex]

∴ [tex]r^3 = 8\\r = 2\\[/tex]

Common ratio = 2

To find the nth number of terms = [tex]\frac{a(r^n -1)}{r-1\\}[/tex]

∴ [tex]\frac{a(2^{10} -1)}{2-1}[/tex]

∴ [tex]1023a\\[/tex]

The sum of 10 terms = 1023a

The sum of 10 terms = 2557.5

∵ 2557.5 = 1023a

∵ [tex]a = \frac{5}{2}[/tex]

To find the 10th term:

∴ [tex]ar^9\\[/tex]

∴ [tex]\frac{5}{2} *2^9[/tex]

⇒ 1280   ║ answer.

Eduard22sly

The 10th term of the sequence is 1280

From the question given above, we were told that the fourth term is 8 times the first term. This can be written as:

T₄ = 8a

But

T₄ = ar³

a => is the first term

r => is the common ratio

Thus,

ar³ = 8a

Solving for the common ratio (r):

ar³ = 8a

Divide both side by a

r³ = 8

Take the cube root of both side

r = ³√8

r = 2

Thus, the common ratio (r) is 2

Next, we shall determine the first term (a). This can be obtained as follow:

Sum of 10th term (S₁₀) = 2557.5

Common ratio (r) = 2

Number of term (n) = 10

First term (a) =?

Sₙ = a[rⁿ – 1] / r – 1

2557.5 = a[2¹⁰ – 1] / 2 – 1

2557.5 = a[1024 – 1]

2557.5 = 1023a

Divide both side by 1023

a = 2557.5 / 1023

a = 2.5

Thus, the first term is 2.5

Finally, we shall determine the 10th term of the sequence. This can be obtained as follow:

Common ratio (r) = 2

First term (a) = 2.5

10th term (T₁₀) =?

T₁₀ = ar⁹

T₁₀ = 2.5 × 2⁹

T₁₀ = 2.5 × 512

T₁₀ = 1280

Therefore, the 10th term of the sequence is 1280

Learn more: https://brainly.com/question/16929076

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