Let's address the problems from the image provided:

Problem 14:

1. Find the geometric series. $6 + p + q + r + 7776 + \ldots$ is a geometric series.

2. Sum of first 10th and 20th terms of an arithmetic series are 155 and 610 respectively.

   • (a) Which term of the series $7 + 13 + 19 + 25 + \ldots$ is 319?
   • (b) Find the value of $p, q,$ and $r$.
   • (c) Find the sum of the first 30th term of the arithmetic series.

Solutions:

Part 1: Find the geometric series.

Given series: $6, p, q, r, 7776, \ldots$

In a geometric series, the ratio between consecutive terms is constant. Let’s denote the common ratio as $r$.

$p = 6r, \quad q = 6r^2, \quad r = 6r^3, \quad 7776 = 6r^4$

From $6r^4 = 7776$, we find:

$r^4 = \frac{7776}{6} = 1296$ $r = \sqrt[4]{1296} = 6$

Hence, the series is $6, 36, 216, 1296, 7776, \ldots$.

Part 2: Sum of the first 10th and 20th terms of an arithmetic series.

The formula for the nth term of an arithmetic sequence is: $a_n = a_1 + (n-1)d$

The sum of the nth term formula is: $S_n = \frac{n}{2} (2a + (n - 1)d)$

Given:

• $a_{10} = 155$
• $a_{20} = 610$

We can set up two equations based on the nth term formula:

1. $a + 9d = 155$
2. $a + 19d = 610$

Solving these simultaneously gives us $d = 45.5$ and $a = -254$.

Part 3: Which term of the series $7 + 13 + 19 + 25 + \ldots$ is 319?

Using the formula for the nth term of an arithmetic sequence $a_n = a + (n - 1)d$.

Here, $a = 7$ and $d = 6$.

$7 + (n - 1) \cdot 6 = 319$ $(n - 1) \cdot 6 = 312$ $n - 1 = 52$ $n = 53$

The 53rd term is 319.

Part 4: Find the sum of the first 30 terms of the arithmetic series.

Sum of the first n terms, $S_n = \frac{n}{2} (2a + (n-1)d)$.

For the first 30 terms: $S_{30} = \frac{30}{2} \times (2 \times 7 + 29 \times 6)$ $S_{30} = 15 \times (14 + 174) = 15 \times 188 = 2820$

Summary of Results:

1. Geometric series: $6, 36, 216, 1296, 7776, \ldots$.
2. Value of $a$ and $d$: $a = -254, d = 45.5$.
3. 53rd term is 319.
4. Sum of the first 30 terms: 2820.

Let me know if you would like more details or have any questions. Here are 5 related questions:

1. How to derive the formula for the nth term of a geometric series?
2. What is the formula for the sum of an infinite geometric series?
3. Can you find the sum of the first n terms for the arithmetic series?
4. How would you find the common ratio if the geometric series started with a different number?
5. What are the properties of arithmetic progressions?

Tip: Always check if the problem involves a finite or infinite geometric series, as this changes the formula used for the sum.