Calculate s35 for the arithmetic sequence in which a5=19 and the common difference is d=-1.3

Answer is 73.5

Work Shown:

a_5 = 19 and d = -1.3 are given values

a_n = a_1 + d(n-1)

a_n = a_1 + (-1.3)(n-1)   replace d with -1.3

a_n = a_1 - 1.3(n-1)

a_5 = a_1 - 1.3(5-1)    replace n with 5

a_5 = a_1 - 1.3(4)

a_5 = a_1 - 5.2

19 = a_1 - 5.2   replace "a_5" with 19; isolate a_1

19+5.2 = a_1 - 5.2 + 5.2

24.2 = a_1

a_1 = 24.2

The first term is 24.2 and the common difference is -1.3, so the nth term formula is

a_n = a_1 + d(n-1)

a_n = 24.2 + (-1.3)(n-1)

a_n = 24.2 - 1.3(n-1)

a_n = 24.2 - 1.3n - 1.3(-1)

a_n = 24.2 - 1.3n + 1.3

a_n = -1.3n + 25.5

note: if you plugged n = 5 into that formula above, you should get a_5 = 19

Use this formula to find the 35th term

a_n = -1.3n + 25.5

a_35 = -1.3*35 + 25.5

a_35 = -20

Now we can use the nth term summation formula for arithmetic sequences

s_n = (n/2)*(a_1 + a_n)

s_35 = (35/2)*(a_1 + a_35)       replace n with 35

s_35 = (35/2)*(24.2 + (-20))

s_35 = 73.5

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