The geometric sequence a_(i) is defined by the formula: {:[a_(1)=8],[a_(i)=a_(i-1)*(-1.5)]:} Find the sum of the first 20 terms in the sequence. Choose 1 answer: (A) -2.47*10^(17) (B) -53,220.11 (C) -17,734.70 (D) -10,637.62

Question

The geometric sequence  a_(i) is defined by the formula:  {:[a_(1)=8],[a_(i)=a_(i-1)*(-1.5)]:} Find the sum of the first 20 terms in the sequence. Choose 1 answer: (A)  -2.47*10^(17) (B)  -53,220.11 (C)  -17,734.70 (D)  -10,637.62

Bytelearn · Accepted Answer

Identify type of sequence: Identify the type of sequence. The sequence is geometric because each term is found by multiplying the previous term by a constant ratio.Determine common ratio: Determine the common ratio (r) of the sequence. The common ratio is given by the formula a_(i) = a_(i-1) * (-1.5), so r = -1.5.Use sum formula for sequence: Use the formula for the sum of the first n terms of a geometric sequence. The sum of the first n terms (S_n) of a geometric sequence is given by S_n = a_1 * (1 - r^n) / (1 - r), where a_1 is the first term, r is the common ratio, and n is the number of terms.Calculate sum of 20 terms: Calculate the sum of the first 20 terms. We have a_1 = 8, r = -1.5, and n = 20. Plugging these values into the formula gives us: S_20 = 8 * (1 - (-1.5)^20) / (1 - (-1.5))Perform necessary calculations: Perform the calculations. First, calculate (-1.5)^20: (-1.5)^20 = 3.35 * 10^14 (approximately) Next, calculate the numerator of the sum formula: 1 - (-1.5)^20 = 1 - 3.35 * 10^14 This results in a negative number, which is expected since the common ratio is negative and greater than 1 in absolute value. Now, calculate the denominator of the sum formula: 1 - (-1.5) = 1 + 1.5 = 2.5 Finally, calculate the sum: S_20 = 8 * (1 - 3.35 * 10^14) / 2.5Complete final calculation: Complete the calculation and find the sum. S_20 = 8 * (-3.35 * 10^14) / 2.5 S_20 = -26.8 * 10^14 / 2.5 S_20 = -10.72 * 10^14 S_20 = -1.072 * 10^16

Full solution

More problems from Identify a sequence as explicit or recursive

Related topics