All the terms of a geometric progression are positive. The second and the fourth terms of the progression are 9.6 and 6.144, respectively. Calculate, correct to three significant f... All the terms of a geometric progression are positive. The second and the fourth terms of the progression are 9.6 and 6.144, respectively. Calculate, correct to three significant figures, the sum of the first ten terms of the progression. Read more

Steps to Solve

1. Identify the terms of the geometric progression (GP)

Let the first term be ( a ) and the common ratio be ( r ). We have:

• The second term: ( a r = 9.6 )
• The fourth term: ( a r^3 = 6.144 )

2. Set up equations

From the equations we identified:

• From the second term: $$ a r = 9.6 \quad (1) $$
• From the fourth term: $$ a r^3 = 6.144 \quad (2) $$

3. Divide the second equation by the first

This helps eliminate ( a ): $$ \frac{a r^3}{a r} = \frac{6.144}{9.6} $$

This simplifies to: $$ r^2 = \frac{6.144}{9.6} $$

Calculating this gives: $$ r^2 = 0.64 $$

Taking the square root of both sides: $$ r = 0.8 $$

4. Substitute ( r ) back to find ( a )

Now we can find ( a ) using equation (1): $$ a (0.8) = 9.6 $$ Thus, $$ a = \frac{9.6}{0.8} = 12 $$

5. Use the formula for the sum of the first n terms of a GP

The formula for the sum ( S_n ) of the first ( n ) terms of a geometric series is: $$ S_n = a \frac{1 - r^n}{1 - r} $$

For the first 10 terms (( n = 10 )): $$ S_{10} = 12 \frac{1 - (0.8)^{10}}{1 - 0.8} $$

6. Calculate ( S_{10} )

First, calculate ( (0.8)^{10} ): $$ (0.8)^{10} \approx 0.1073741824 $$

So, $$ S_{10} = 12 \frac{1 - 0.1073741824}{0.2} $$

Now simplifying: $$ S_{10} = 12 \frac{0.8926258176}{0.2} = 12 \times 4.463129088 = 53.657549056 $$

7. Round to three significant figures

Therefore, rounding gives: $$ S_{10} \approx 53.7 $$