Question

Find S₇ for the sequence given by aₙ=1/16(-2)ⁿ-¹

Answer 1

The S₇ for the sequence given by aₙ=1/16(-2)ⁿ-¹ is
`\[S₇ = (1)/(2048)\]`

Step-by-step explanation:

The given sequence is defined by the formula
`\(aₙ = (1)/(16)(-2)^(n-1)\).`To find S₇, the sum of the first seven terms of the sequence, we substitute n = 1, 2, ..., 7 into the formula and add the terms together.

`\[S₇ = a₁ + a₂ + a₃ + a₄ + a₅ + a₆ + a₇\]`

Substituting the values for each term:

`\[S₇ = (1)/(16)(-2)^(0) + (1)/(16)(-2)^(1) + (1)/(16)(-2)^(2) + (1)/(16)(-2)^(3) + (1)/(16)(-2)^(4) + (1)/(16)(-2)^(5) + (1)/(16)(-2)^(6)\]`

Simplifying each term:

`\[S₇ = (1)/(16) + (1)/(-16) + (1)/(64) + (1)/(-128) + (1)/(256) + (1)/(-512) + (1)/(1024)\]`

Combining the terms:

`\[S₇ = (1)/(2048)\]`

Therefore, the sum S₇ for the given sequence is
`\((1)/(2048)\).`

This result can be obtained by recognizing the geometric progression with a common ratio of -2. The formula for the sum of the first n terms of a geometric progression is
`\(Sₙ = (a(1-r^n))/(1-r)\)`, where a is the first term and r is the common ratio. In this case,
`\(a = (1)/(16)\)` and r = -2. Substituting these values into the formula, we get
`\(S₇ = (1)/(2048)\),` confirming the result obtained through direct summation.