Write the related series for the finite sequence 3, 7, 11, 27. then find the sum

The sum =105

Step-by-step explanation:

It is given that, finite sequence 3, 7, 11, ..., 27.

Points to remember

1). For a finite sequence sum of n term is given by

Sum  = n/2[first term + last term]

2). nth term of of an AP

tₙ = a + (n - 1)d

Where a - First term

d - common difference

To find the number of terms in the given sequence

Here a = 3 and d = 7 - 3 = 4

tₙ = a + (n - 1)d

27  = 3 + (n - 1)*4

4(n -1) = 27 - 3 = 24

n - 1 = 24/4 = 6

n = 6 +1 =7

To find the sum

Sum  = n/2[first term + last term]

 = 7/2[3 + 27]

 = 7/2[30] = 7 * 15

 = 105

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.