d) an — — 4 аn-1 + 2n + 3, ао
Permutations and Combinations
If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.
Counting Theory
The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.
![**Problem 31:** What is the value of each of these sums of terms of a geometric progression?
### d) \(\sum_{j=0}^{8} 2 \cdot (-3)^j\)
In this problem, we are asked to find the value of the sum of the first 9 terms (from \( j = 0 \) to \( j = 8 \)) of a geometric progression, where each term is given by \( 2 \cdot (-3)^j \).
To find the value of this series, we can use the formula for the sum of the first \( n \) terms of a geometric progression:
\[ S_n = a \cdot \frac{1 - r^n}{1 - r} \]
Where:
- \( S_n \) is the sum of the first \( n \) terms.
- \( a \) is the first term of the progression.
- \( r \) is the common ratio.
- \( n \) is the number of terms.
For this series:
- \( a = 2 \) (the first term when \( j = 0 \)).
- \( r = -3 \) (the common ratio).
- The series has 9 terms (\( n = 9 \)).
Therefore, substituting these values into the formula:
\[ S_9 = 2 \cdot \frac{1 - (-3)^9}{1 - (-3)} \]
\[ S_9 = 2 \cdot \frac{1 - (-19683)}{1 + 3} \]
\[ S_9 = 2 \cdot \frac{1 + 19683}{4} \]
\[ S_9 = 2 \cdot \frac{19684}{4} \]
\[ S_9 = 2 \cdot 4921 \]
\[ S_9 = 9842 \]
Thus, the value of the sum of the given geometric progression is **9842**.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1a03f387-3c7c-47a4-96eb-2cc1ae2384ed%2F4015eb4e-9ef6-4570-b2cc-dd05ace89200%2Frybhyph_processed.png&w=3840&q=75)
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