Question

A geometric series has a common ratio of -3. Given that the sum of the first nine terms of the progression is 703, find the first term.

Answer 1

To determine the first term of a geometric series with a common ratio of -3 and the sum of the first nine terms as 703, we use the series sum formula. By substituting the known values into the formula, we calculate that the first term (a1) is 1.

Step-by-step explanation:

The question asks to find the first term of a geometric series with a common ratio of -3 and a sum of the first nine terms equal to 703. To solve this, we can use the formula for the sum of the first n terms of a geometric series, which is Sn = a1(1 - rn) / (1 - r), where Sn is the sum of the first n terms, a1 is the first term, and r is the common ratio. Since we know that S9 = 703 and r = -3, we can substitute these values into the formula and solve for a1.

To find a1:

1. 
   
3. Substitute S9 = 703 and r = -3 into the sum formula: 703 = a1(1 - (-3)9) / (1 - (-3)).
4. 
   
6. Calculate the numerator: 1 - (-3)9 = 1 - (-19683).
7. 
   
9. Since -3 is negative, and the exponent is odd, the result is also negative, which then becomes 1 - (-19683) = 19684.
10. 
    
12. Now calculate the denominator: 1 - (-3) = 1 + 3 = 4.
13. 
    
15. Divide both sides of the equation by 4 to isolate a1: 703 = a1 * 19684 / 4.
16. 
    
18. Finally, solve for a1: a1 = 703 * 4 / 19684.
19. 
    
21. After performing the division, a1 = 1.

Therefore, the first term of the geometric series is 1.