Question

What is the sum of the first 6 terms in this series?

6 + (-18) + 54 + (-162) ...
-1,092
-2,184
1,092
2,184

Answer 1

Step-by-step explanation

Compute the ratios of all the adjacent terms:

`r=(a_(n+1))/(a_n)`
`(-18)/(6)=-3,\: \quad (54)/(-18)=-3,\: \quad (-162)/(54)=-3`
`\mathrm{The\: ratio\: of\: all\: the\: adjacent\: terms\: is\: the\: same\: and\: equal\: to}`
`r=-3`
`a_n=a_1r^{\mleft\{n-1\mright\}}`
`\mathrm{For\: }a_1=6,\: r=-3`
`a_n=6\mleft(-3\mright)^(n-1)`

Now, plugging in the term 5 into the equation:

`a_n=6\cdot(-3)^(5-1)`

Subtracting numbers:

`a_n=6\cdot(-3)^4=6\cdot81=486`

The 5th term is 486.

Now, we need to compute the 6th term:

`a_n=6\cdot(-3)^(6-1)=6\cdot(-3)^5`

Computing the numbers:

`a_n=6\cdot(-243)`

Multiplying the numbers:

`a_n=-1,458`

The term 6th is -1458

Adding the first 6 terms:

6 + (-18) + 54 + (-162) + 486 + (-1,458) = -1,092

In conclusion, the solution is -1,092