Question

Find the sum of the first seven terms of the geometric sequence series having the following information: 2-8+32- … +a n

Answer 1

`S_7=6554`

Explanation:

`\boxed{\begin{minipage}{7 cm}\underline{Sum of the first $n$ terms of a geometric series}\\\\$S_n=(a(1-r^n))/(1-r)$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the first term. \\ \phantom{ww}$\bullet$ $r$ is the common ratio.\\\end{minipage}}`

Given geometric series:

`2-8+32-...+a_n`

The first term (a) of the sequence is 2:

`\implies a=2`

To find the common ratio (r), divide consecutive terms:

`\implies r=(a_2)/(a_1)=(-8)/(2)=-4`

To find the sum of the first 7 terms, substitute the found values of a and r into the formula, along with n = 7:

`\implies S_7=(2(1-(-4)^7))/(1-(-4))`

`\implies S_7=(2(1-(-16384)))/(5)`

`\implies S_7=(2(16385))/(5)`

`\implies S_7=(32770)/(5)`

`\implies S_7=6554`

Answer 2

• 6554

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From the given information we can see:

• The first term is a = 2,
• Common ratio is r = - 4

Use the equation for the sum of the first n terms:

• Sₙ = a(rⁿ - 1)/(r - 1)
• S₇ = 2((-4)⁷ - 1)/(-4 - 1) = 6554