Question

Given a term in an arithmetic sequence and the common difference find the first five terms and the explicit formula. a

36
​
=−276;d=−7 5. Evaluate the following using the finite geometric sum formula. ∑
k=1
7
​
4

Answer 1

To find the first terms and the explicit formula of an arithmetic sequence, one must use the arithmetic sequence formula, incorporating the given term and common difference to solve for the first term. The first term is then used to calculate the subsequent terms.

Step-by-step explanation:

Problem 1: Arithmetic Sequence

Given:
`a_{36` = -276, d = -7

To find: First five terms and explicit formula

Solution:

Find the first term (
`a_1`):

`a_1` =
`a_{36` - (36-1) × d = -276 - 35 × (-7) = -24

Find the first five terms:

`a_1` = -24

`a_2` =
`a_1` + d = -31

`a_3` =
`a_2` + d = -38

`a_4` =
`a_3` + d = -45

`a_5` =
`a_4` + d = -52

First five terms: -24, -31, -38, -45, -52

Explicit formula:

`a_n` =
`a_1` + (n-1) × d

`a_n` = -24 - 7(n-1)

Problem 2: Geometric Series

Given: ∑(k=1 to 7) 4

To evaluate: The sum using the finite geometric sum formula

Solution:

Identify the values:

a = the first term = 4

r = the common ratio = 4 (since each term is 4 times the previous term)

n = the number of terms = 7

Apply the finite geometric sum formula:

`S_n` = a × (1 - rⁿ) / (1 - r)

`S_7` = 4 × (1 - 4⁷) / (1 - 4) = 4 × (1 - 16384) / (-3) = 21844

Therefore, the sum of the series is 21844.

Answer 2

To find the first five terms of an arithmetic sequence and the explicit formula, we can use the given term and the common difference. Evaluating the finite geometric sum formula, we can find the sum of seven 4s. So, the solution is 28.

Step-by-step explanation:

To find the first five terms of an arithmetic sequence, we use the formula:

a_n = a_1 + (n - 1)d

Given the term a_1 = 36 and the common difference d = -7, we can substitute these values into the formula. Evaluating the formula for the first five terms:

a_1 = 36

a_2 = 36 + (-7) = 29

a_3 = 29 + (-7) = 22

a_4 = 22 + (-7) = 15

a_5 = 15 + (-7) = 8

The explicit formula for the arithmetic sequence can be derived by rearranging the formula:

a_n = a_1 + (n - 1)d

to:

a_n = 36 + (n - 1)(-7)

The given finite geometric sum formula is:

∑k=17 4 = 4 + 4 + 4 + 4 + 4 + 4 + 4 = 28