Find sn for the arithmetic series 5+7+9 + … and determine the value of n for which the series has sum 165.

The sum of n terms for the arithmetic series is:

The value of n for which the series has sum 165 is 11

The given series is:

5 + 7 + 9 +.......

The first value, a = 5

The common difference, d = 7 - 5

d  =  2

The sum of the first n terms of an arithmetic series is given by the formula:

Substituting a = 5, and d = 2 into the sum of the series, we have:

To determine the value of n for which the series has sum 165, substitute into the sum above

The value of n for which the series has sum 165 is 11

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see explanation

Step-by-step explanation:

the sum to n terms of an arithmetic sequence is

= [2a + (n - 1)d ]

where d is the common difference and a is the first term

here d = 9 - 7 = 7 - 5 = 2 and a = 5, hence

= [(2 × 5) + 2(n - 1) ]

                        = (10 + 2n - 2)

                        = (2n + 8)

                        = n² + 4n

When sum = 165, then

n² + 4n = 165 ← rearrange into standard form

n² + 4n - 165 = 0 ← in standard form

(n + 15)(n - 11) = 0 ← in factored form

equate each factor to zero and solve for n

n + 15 = 0 ⇒ n = - 15

n - 11 = 0 ⇒ n = 11

but n > 0 ⇒ n = 11