Question

Evaluate the series 2 8 32 128 512 2048.

Answer 1

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,
`a_n` denote the n-th term of the given series.

Then, we see the following pattern in the consecutive terms of the given series:

`a_1= 2, \\\\a_2=8=2* 4=4a_1, \\\\a_3=32=8* 4=4a_2\\\\a_4=128=32*4=4a_3,\\\\a_5=2048=128*4=4a_4,\\\\\vdots~~~~~~\vdots~~~~~~\vdots`

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

`S_5\\\\=2+8+32+128+512+2048\\\\=(a(r^6-1))/(r-1)\\\\\\=(2(4^6-1))/(4-1)\\\\\\=(2)/(3)* (4096-1)\\\\=2*1365\\\\=2730.`

The required sum of the series is 2730.