Show work and explain with formulas.

7. find the sum: a_1 = 91, d = -4, a_n = 15

8. find the ninth term of a geometric sequence in which a_3 = 63 and r = -3

9. find the sum of geometric series for which a_1 = 729, a_n = -3, r = -1/3

Q7. 945; Q8. 45 927; Q9. 546.75

Step-by-step explanation:

7. Arithmetic sequence

The explicit formula for the nth term of an arithmetic sequence is

aₙ = a₁ + d(n - 1 )

a₁ is the first term, and d is the difference in value between consecutive terms. Thus,  

a₁ = 91, d = -4, n = 15

a₁₅ = 91 - 4(15 - 1) = 91 - 4(14) = 91 - 56 = 35

We find the sum of an arithmetic series by multiplying the number of terms by the average of the first and last terms.  

S = n[(a₁ + a₁₅)/2]

S = 15[(91 + 35)/2] = 15 × 126/2 = 945

The sum of the first 15 terms is 945.

8. nth term of geometric sequence

a₃ = 63 and r = -3

The explicit formula for the nth term of a geometric sequence is

aₙ = a₁rⁿ⁻¹

a₁ is the first term and r is the common ratio.

If we start counting from a₃, then a₉ is the seventh term in the sequence.

In your sequence, r = -3.

a₇ = 63(-3)⁷⁻¹ = 63(-3)⁶ = 63 × 729 = 45 927

The ninth term in the sequence is 45 927.

9. Sum of geometric series

a₁ = 729, aₙ = -3, r = -⅓

I don't know what you mean by aₙ = -3. It says that every term is -3, so I am going to ignore it.

Since |r| <1, we have a convergent series, and the formula for the sum is

S = a₁/(1 - r)

∴ S = 729/[(1 - (-⅓)] = 729/1⅓ = 729/(⁴/₃) = 729 × ¾ = 546.75

The sum of the geometric series is 546.75.

7 1060

Step-by-step explanation:

8 45,927

Step-by-step explanation:

9 546

Step-by-step explanation:

18

step-by-step explanation:

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