Question

В геометрической прогрессии найти: 2) b1 и b8,Если q=-2 , S8=85

Answer 1

Task/17023403
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В геометрической прогрессии найти:  b₁ и b₈ ,если q=-2 , S₈=85
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S₈ =b₁(1-q⁸)/(1-q) ;
85 =b₁(1-(-2)⁸)/(1-(-2) ;
85 =-(255/ 3 ) *b₁ ;
b₁ = -1.
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b₈ =b₁q⁷ 
b₈ =(-1)*(-2)⁷ =128.

ответ:  b₁ = -1 ; b₈ =128.

Answer 2

Finding the First Term (b1) and the Eighth Term (b8) in a Geometric Progression

In a geometric progression, each term is found by multiplying the previous term by a constant ratio. The formula for finding the nth term of a geometric progression is:

bn = b1 * q^(n-1)

where bn represents the nth term, b1 is the first term, q is the common ratio, and n is the position of the term.

In this case, we are given the following information:

- q = -2 (the common ratio) - S8 = 85 (the sum of the first 8 terms)

To find the first term (b1) and the eighth term (b8), we need to use the formula for the nth term and the formula for the sum of a geometric progression.

Finding the First Term (b1)

To find the first term (b1), we can use the formula for the sum of a geometric progression:

Sn = b1 * (1 - q^n) / (1 - q)

where Sn represents the sum of the first n terms.

In this case, we are given that S8 = 85. Substituting this into the formula, we have:

85 = b1 * (1 - (-2)^8) / (1 - (-2))

Simplifying the equation, we get:

85 = b1 * (1 - 256) / 3

Multiplying both sides by 3, we have:

255 = b1 * (-255)

Dividing both sides by -255, we find:

b1 = -1

Therefore, the first term (b1) in the geometric progression is -1.

Finding the Eighth Term (b8)

Now that we know the first term (b1) is -1, we can use the formula for the nth term to find the eighth term (b8).

b8 = b1 * q^(8-1)

Substituting the given value of q = -2, we have:

b8 = -1 * (-2)^(8-1)

Simplifying the equation, we get:

b8 = -1 * (-2)^7

Evaluating the expression, we have:

b8 = -1 * (-128)

b8 = 128

Therefore, the eighth term (b8) in the geometric progression is 128.

To summarize, in the given geometric progression with a common ratio (q) of -2 and a sum of the first 8 terms (S8) equal to 85, the first term (b1) is -1 and the eighth term (b8) is 128.