Find the sum of the first 7 terms of the following sequence. Round to the nearest hundredth if necessary. 6, comma, minus, 10, comma, start fraction, 50, divided by, 3, end fraction, comma, dot dot dot 6,−10, 3 50 ​ ,... Sum of a finite geometric series: Sum of a finite geometric series: S, start subscript, n, end subscript, equals, start fraction, a, start subscript, 1, end subscript, minus, a, start subscript, 1, end subscript, r, to the power n , divided by, 1, minus, r, end fraction S n ​

The sequence you provided is a geometric sequence:

$6, -10, \frac{50}{3}, \dots$

Step 1: Identify the first term and common ratio.

• First term $a_1 = 6$
• To find the common ratio $r$, divide the second term by the first term:

$r = \frac{-10}{6} = -\frac{5}{3}$

Step 2: Use the formula for the sum of the first $n$ terms of a geometric series:

The sum of the first $n$ terms of a geometric series is given by:

$S_n = \frac{a_1(1 - r^n)}{1 - r}$

For this sequence, we need to find the sum of the first 7 terms, so $n = 7$, $a_1 = 6$, and $r = -\frac{5}{3}$.

Step 3: Calculate $S_7$:

$S_7 = \frac{6 \left( 1 - \left( -\frac{5}{3} \right)^7 \right)}{1 - \left( -\frac{5}{3} \right)}$

Step 4: Compute powers and simplify.

1. Calculate $\left( -\frac{5}{3} \right)^7$.

   • This gives $\left( -\frac{5}{3} \right)^7 = -\frac{78125}{2187}$.

2. Now, substitute this into the sum formula:

$S_7 = \frac{6 \left( 1 - \left( -\frac{78125}{2187} \right) \right)}{1 + \frac{5}{3}}$

Simplifying the denominator $1 + \frac{5}{3} = \frac{8}{3}$.

$S_7 = \frac{6 \left( 1 + \frac{78125}{2187} \right)}{\frac{8}{3}} = \frac{6 \left( \frac{2187 + 78125}{2187} \right)}{\frac{8}{3}}$

Simplifying:

$S_7 = \frac{6 \times \frac{80312}{2187}}{\frac{8}{3}} = \frac{6 \times 80312}{2187} \times \frac{3}{8}$

$S_7 = \frac{6 \times 80312 \times 3}{2187 \times 8} = \frac{1445616}{17496} \approx 82.7$

Final Answer:

The sum of the first 7 terms is approximately 82.7.

Would you like to go over any part of the calculation in more detail or explore further examples?

---

Additional Questions:

1. How does the common ratio affect the sum of a geometric series?
2. Can you find the sum of the first 10 terms of a different geometric sequence?
3. How do you determine whether a geometric series converges or diverges?
4. What happens to the sum of a geometric series when the common ratio is between -1 and 1?
5. How do you calculate the infinite sum of a convergent geometric series?

Tip: Always check the common ratio $r$ to ensure that it’s not equal to 1, as the sum formula won't work when $r = 1$.