Respuesta :
There are 9 terms in geometric series 8 + 40 + 200 + ... + 3,125,000
Solution:
Need to determine how nay terms are there in following geometric series
8 + 40 + 200 + ... + 3,125,000
Let’s derived basic properties of given geometric series which will be helpful in evaluating number of terms
[tex]\begin{array}{l}{\text { First term of given geometric series is } a_{1}=8} \\\\ {\text { Second term of given geometric series is } a_{2}=40} \\\\ {\text { Third term of given geometric series is } a_{3}=200}\end{array}[/tex]
[tex]\text { Common ratio of geometric series } \mathrm{r}=\frac{a_{2}}{a_{1}}=\frac{a_{3}}{a_{2}}=5[/tex]
Formula for nth term of geometric series is as follows
[tex]a_{n}=a_{1} r^{(n-1)}[/tex]
As last term is 3,125,000
[tex]\text { So } a_{n}=3,125,000[/tex]
On substituting value of [tex]a_1, a_n[/tex] and r is formula of nth term of geometric series we get
[tex]\begin{array}{l}{3,125,000=8 \times 5^{(n-1)}} \\\\ {=>\frac{3125000}{8}=5^{(n-1)}} \\\\ {=>390625=5(n-1)} \\\\ {=>5^{8}=5^{(n-1)}}\end{array}[/tex]
Since base that is 5 is same on both sides, so exponent will be equal
=>8 = n – 1
=> n = 8 + 1 = 9
=> n = 9
Hence there are 9 terms in geometric series
