Let's analyze the given geometric series and solve the problem step-by-step.
### a) Given geometric series: [tex]\( 25 + 50 + 100 + \ldots \)[/tex]
#### (i) If the sum of the terms of the series is 775, how many terms are there?
To solve this, we need to use the formula for the sum of the first [tex]\( n \)[/tex] terms of a geometric series:
[tex]\[ S_n = a \frac{r^n - 1}{r - 1} \][/tex]
where:
- [tex]\( S_n \)[/tex] is the sum of the first [tex]\( n \)[/tex] terms,
- [tex]\( a \)[/tex] is the first term,
- [tex]\( r \)[/tex] is the common ratio,
- [tex]\( n \)[/tex] is the number of terms.
From the series [tex]\( 25 + 50 + 100 + \ldots \)[/tex]:
- The first term [tex]\( a = 25 \)[/tex],
- The common ratio [tex]\( r = 2 \)[/tex].
We are given that the sum of the terms is [tex]\( S_n = 775 \)[/tex].
Using the sum formula:
[tex]\[ 775 = 25 \frac{2^n - 1}{2 - 1} \][/tex]
[tex]\[ 775 = 25 (2^n - 1) \][/tex]
[tex]\[ 775 = 25 \cdot 2^n - 25 \][/tex]
[tex]\[ 775 + 25 = 25 \cdot 2^n \][/tex]
[tex]\[ 800 = 25 \cdot 2^n \][/tex]
[tex]\[ 2^n = \frac{800}{25} \][/tex]
[tex]\[ 2^n = 32 \][/tex]
[tex]\[ 2^n = 2^5 \][/tex]
Thus, [tex]\( n = 5 \)[/tex].
So, the number of terms is [tex]\( \boxed{5} \)[/tex].
#### (ii) What is the last term of the series?
To find the last term of the series [tex]\( T_n \)[/tex], we use the formula for the [tex]\( n \)[/tex]-th term of a geometric series:
[tex]\[ T_n = a \cdot r^{n-1} \][/tex]
From the previous part, we know:
- The first term [tex]\( a = 25 \)[/tex],
- The common ratio [tex]\( r = 2 \)[/tex],
- The number of terms [tex]\( n = 5 \)[/tex].
Using the formula:
[tex]\[ T_5 = 25 \cdot 2^{5-1} \][/tex]
[tex]\[ T_5 = 25 \cdot 2^4 \][/tex]
[tex]\[ T_5 = 25 \cdot 16 \][/tex]
[tex]\[ T_5 = 400 \][/tex]
So, the last term of the series is [tex]\( \boxed{400} \)[/tex].
