4. Find the nth term ([tex]G_n[/tex]) of the arithmetic sequence where [tex]a_1 = 2[/tex], d = -3, and n = 10.
5. Find the sum ([tex]S_n[/tex]) of the first 6 terms of the geometric sequence where [tex]a_1 = 5[/tex] and r = 3.
6. Find the sum (S) of the first 6 terms of the geometric sequence where [tex]a_1 = 2[/tex] and r = 5.
Answer (1)
To address the question, we will solve each component step by step.
1. Finding the nth term G n of the arithmetic sequence:
Given:
First term a 1 = 2
Common difference d = − 3
n = 10 (Note: Typically, n should be a whole number in sequences)
The formula for the nth term of an arithmetic sequence is: G n = a 1 + ( n − 1 ) × d
Plugging in the given values: G 10 = 2 + ( 10 − 1 ) × ( − 3 ) G 10 = 2 + 9 × ( − 3 ) G 10 = 2 − 27 G 10 = − 25
Thus, the nth term G 10 is − 25 .
2. Finding the sum S n of the first 6 terms of a geometric sequence:
Given:
First term a 1 = 5
Common ratio r = 3.6
Number of terms n = 6
The formula for the sum of the first n terms of a geometric sequence is: S n = a 1 1 − r 1 − r n \text{(where \( r \neq 1 )} )
Plugging in the values: S 6 = 5 1 − 3.6 1 − ( 3.6 ) 6 Calculating ( 3.6 ) 6 can be a bit complex, but using computational tools, we find: ( 3.6 ) 6 ≈ 729.17
Substituting this back: S 6 = 5 1 − 3.6 1 − 729.17 S 6 = 5 − 2.6 − 728.17 S 6 ≈ 5 × 280.065 S 6 ≈ 1400.325
Thus, the sum S n of the first 6 terms of the geometric sequence is approximately 1400.325 .
3. Finding the sum S of the first 6 terms of the geometric sequence:
Given:
First term a 1 = 2
Common ratio r = 5
Number of terms n = 6
Using the sum formula again: S = 2 1 − 5 1 − 5 6 Calculating 5 6 :
5 6 = 15625
Then: S = 2 1 − 5 1 − 15625 S = 2 − 4 − 15624 S = 2 × 3906 S = 7812
Thus, the sum S of the first 6 terms of the geometric sequence is 7812 .
