Certainly! Let's solve the questions step by step. We are given that the fifth term of a geometric sequence is 4375 and the second term is 35.
For a geometric sequence, the [tex]\( n \)[/tex]-th term is given by:
[tex]\[ a_n = a \cdot r^{n-1} \][/tex]
where [tex]\( a \)[/tex] is the first term and [tex]\( r \)[/tex] is the common ratio.
### (i) Find the third term
We know:
- The fifth term ([tex]\( a \cdot r^4 \)[/tex]) is 4375.
- The second term ([tex]\( a \cdot r \)[/tex]) is 35.
To find the values of [tex]\( a \)[/tex] and [tex]\( r \)[/tex]:
[tex]\[ a \cdot r = 35 \][/tex]
[tex]\[ a \cdot r^4 = 4375 \][/tex]
From these two equations, we can solve for [tex]\( r \)[/tex]:
[tex]\[ \frac{a \cdot r^4}{a \cdot r} = \frac{4375}{35} \][/tex]
[tex]\[ r^3 = \frac{4375}{35} \][/tex]
[tex]\[ r^3 = 125 \][/tex]
[tex]\[ r = 5 \][/tex]
Now substituting [tex]\( r = 5 \)[/tex] back into the equation for the second term to find [tex]\( a \)[/tex]:
[tex]\[ a \cdot 5 = 35 \][/tex]
[tex]\[ a = 7 \][/tex]
With [tex]\( a \)[/tex] and [tex]\( r \)[/tex] known, the third term ([tex]\( a \cdot r^2 \)[/tex]) can be found:
[tex]\[ a_3 = a \cdot r^2 \][/tex]
[tex]\[ a_3 = 7 \cdot 5^2 \][/tex]
[tex]\[ a_3 = 7 \cdot 25 \][/tex]
[tex]\[ a_3 = 175 \][/tex]
So, the third term is [tex]\( 175 \)[/tex].
### (ii) Find the sixth term
The sixth term is given by:
[tex]\[ a_6 = a \cdot r^5 \][/tex]
[tex]\[ a_6 = 7 \cdot 5^5 \][/tex]
[tex]\[ a_6 = 7 \cdot 3125 \][/tex]
[tex]\[ a_6 = 21875 \][/tex]
So, the sixth term is [tex]\( 21875 \)[/tex].
### (iii) Find the sum of the first five terms
The sum of the first [tex]\( n \)[/tex] terms of a geometric sequence is given by:
[tex]\[ S_n = a \cdot \frac{r^n - 1}{r - 1} \][/tex]
For the first five terms ([tex]\( n = 5 \)[/tex]):
[tex]\[ S_5 = 7 \cdot \frac{5^5 - 1}{5 - 1} \][/tex]
[tex]\[ S_5 = 7 \cdot \frac{3125 - 1}{4} \][/tex]
[tex]\[ S_5 = 7 \cdot \frac{3124}{4} \][/tex]
[tex]\[ S_5 = 7 \cdot 781 \][/tex]
[tex]\[ S_5 = 5467 \][/tex]
So, the sum of the first five terms is [tex]\( 5467 \)[/tex].
### Summary
1. The third term is [tex]\( 175 \)[/tex].
2. The sixth term is [tex]\( 21875 \)[/tex].
3. The sum of the first five terms is [tex]\( 5467 \)[/tex].
