summations discrete math

Question

Find the value of each of the sum.

7

sigma

j=1

(5.3^j−1 − 3.5^j )

Answer 1

Insert the 1^st value of "j" into the expression to get the 1^st term:

5.3^{(1 - 1)} - 3.5¹ = 5.3⁰ - 3,5 = 1 - 3.5 = -2.5

Insert the 2^nd value of "j" into the expression to get the 2^nd term:

5.3^{(2 - 1)} - 3.5² = 5.3¹ - 12.25 = 5.3 - 12.25 = -6.95

The sum now looks like this: -2.5 + -6.95

Insert the 3^rd value of "j" into the expression to get the 3^rd term:

5.3^{(3 - 1)} - 3.5³ = 5.3² - 42.875 = 28.09 - 42.875 = -14.785

The sum now looks like this: -2.5 + -6.95 + -14.785

Insert the 4^th value of "j" into the expression to get the 4^th term:

5.3^{(4 - 1)} - 3.5⁴ = 5.3³ - 150.0625 = 148.877 - 150.0625 = -1.1855

The sum now looks like this: -2.5 + -6.95 + -14.785 + -1.1855

Insert the 5^th value of "j" into the expression to get the 5^th term:

5.3^{(5 - 1)} - 3.5⁵ = 5.3⁴ - 525.21875 = 789.0481 - 525.21875 = 263.82935

The sum now looks like this: -2.5 + -6.95 + -14.785 + -1.1855 + 263.82935

Insert the 6^th value of "j" into the expression to get the 6^th term:

5.3^{(6 - 1)} - 3.5⁶ = 5.3⁵ - 1838.265625 = 4181.95493 - 1838.265625 = 2343.689305

The sum now looks like this: -2.5 + -6.95 + -14.785 + -1.1855 + 263.82935 + 2343.689305

Insert the 7^th value of "j" into the expression to get the 7^th term:

5.3^{(7 - 1)} - 3.5⁷ = 5.3⁶ - 6433.929688 = 22164.36113 - 6433.929688 = 15730.43144

The sum now looks like this: -2.5 + -6.95 + -14.785 + -1.1855 + 263.82935 + 2343.689305 + 15730.43144

Adding these values all up you get 18312.529595

If you have a TI-84 Plus CE, you can use the math button, scroll down to "summation ∑(" and select it. You can then type in the equation as it is written and the calculator will give you the answer.