Answer :

Answer:

4.

Step-by-step explanation:

To find the number of terms in a geometric progression where the sum [tex] ( S_n) [/tex] is given, we need to use the formula for the sum of a geometric series. Let's break it down step-by-step:

1. Identify the first term and common ratio:

The given series is [tex] 0.5 - 0.1 + 0.02 - \ldots [/tex] .

  • The first term (a) is 0.5.
  • The common ratio (r) can be found by dividing the second term by the first term:

[tex] r = \dfrac{-0.1}{0.5} = -0.2 [/tex]

2. Sum of the first n terms of a geometric series:

The sum [tex] S_n [/tex] of the first n terms of a geometric series with common ratio r is given by:

[tex]S_n = a \dfrac{1 - r^n}{1 - r} [/tex]

Since we know [tex] S_n = 0.416 [/tex], [tex] a = 0.5 [/tex], and [tex]r = -0.2 [/tex], we can plug these values into the formula:

[tex] 0.416 = 0.5 \dfrac{1 - (-0.2)^n}{1 - (-0.2)} [/tex]

3. Simplify the equation:

[tex]0.416 = 0.5 \dfrac{1 - (-0.2)^n}{1 + 0.2} [/tex]

[tex] 0.416 = 0.5 \dfrac{1 - (-0.2)^n}{1.2} [/tex]

[tex] 0.416 = \dfrac{0.5}{1.2} (1 - (-0.2)^n) [/tex]

[tex] 0.416 = \dfrac{0.5}{1.2} (1 - (-0.2)^n) [/tex]

Simplify the fraction:

[tex] 0.416 = \dfrac{5}{12} (1 - (-0.2)^n) [/tex]

4. Isolate the exponential term:

Multiply both sides by [tex] \dfrac{12}{5} [/tex]:

[tex] 0.416 \times \dfrac{12}{5} = 1 - (-0.2)^n [/tex]

[tex] 0.9984 = 1 - (-0.2)^n [/tex]

[tex] (-0.2)^n = 1 - 0.9984 [/tex]

[tex] (-0.2)^n = 0.0016 [/tex]

5. Solve for n:

Take the natural logarithm of both sides:

[tex] \ln((-0.2)^n) = \ln(0.0016) [/tex]

Use the property of logarithms [tex][\ln(a^b) = b \ln(a) [/tex]:

[tex] n \ln(-0.2) = \ln(0.0016) [/tex]

Since [tex] \ln(-0.2) [/tex] is undefined for real numbers, we consider the absolute values and use the logarithm properties for negative base in terms of complex numbers. Here we'll use:

[tex] n \ln(0.2) = \ln(0.0016) [/tex]

Calculate the logarithms:

[tex] \ln(0.2) \approx -1.6094 [/tex]

[tex]\ln(0.0016) \approx -6.4378 [/tex]

Now solve for n:

[tex] n = \dfrac{\ln(0.0016)}{\ln(0.2)} [/tex]

[tex] n = \dfrac{-6.4378}{-1.6094} [/tex]

[tex]n \approx 4 [/tex]

So, the number of terms in the series is approximately 4.