Respuesta :
Answer: The given series is a geometric series sum 2730.
Step-by-step explanation: We are given to evaluate the following series :
2, 8, 32, 128, 512, 2048.
Let, [tex]a_n[/tex] denote the n-th term of the given series.
Then, we see the following pattern in the consecutive terms of the given series:
[tex]a_1= 2, \\\\a_2=8=2\times 4=4a_1, \\\\a_3=32=8\times 4=4a_2\\\\a_4=128=32\times4=4a_3,\\\\a_5=2048=128\times4=4a_4,\\\\\vdots~~~~~~\vdots~~~~~~\vdots[/tex]
Therefore, each term after the first one is the product of the preceding term and 4.
That is, the given series is a geometric series with first term 2 and common ratio 4.
Thus, the required sum of the given series is
[tex]S_5\\\\=2+8+32+128+512+2048\\\\=\dfrac{a(r^6-1)}{r-1}\\\\\\=\dfrac{2(4^6-1)}{4-1}\\\\\\=\dfrac{2}{3}\times (4096-1)\\\\=2\times1365\\\\=2730.[/tex]
The required sum of the series is 2730.
