Question

Determine whether each series is arithmetic or geometric. Then evaluate the finite series for the specified term number. 2+6+18+ . . . . . . ; n=6

Answer 1

The given series is a geometric series. The sum of its first 6 terms is 728.

To determine whether the series is an arithmetic or geometric one, we need to check how the consecutive terms relate to each other.

If it is an arithmetic series, then:

  a(n) = a(n-1) + d

where d is the common difference.

If it is a geometric series, then:

  a(n) = a(n-1) . r

where r is the common ratio.

The given series is:

     2+6+18+ . . . .

and the number of terms is 6.

In the given series:

a(1) = 2

a(2) = 6

a(3) = 18

Notice that

a(2)/a(1) = 6/2 = 3

a(3)/a(2) = 18/6 = 3

Since the ratio of consecutive terms are constant, then the given series is a geometric series. Furthermore, its ratio, r = 3

Sum formula for the first n term of a geometric series is:

S(n) = a(1) .  (rⁿ -1)/(r -1)

Substitute n = 6, a(1) = 2, and r = 3

S(6) = 2 . (3⁶ - 1)/(3 - 1)

S(6) = 2. 364

      = 728

We can also evaluate the sum directly. Write all the terms until n = 6.

a(4) = 18 . 3 = 54

a(5) = 54 . 3 = 162

a(6) = 162 . 3 = 486

Hence, the series is:

2+6+18+54+162+486 = 728

Learn more about series here:

brainly.com/question/28108001

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