This is a three part question Please show work Thank you Wri

This is a three part question. Please show work. Thank you.

Write the expression we could use for the nth term of the arithmetic sequence -19, -14, -9, -4, 1,.... Use that equation to find a_36 and S_36. To write the expression we could use for the nth term of the geometric sequence 3,12: 48, 192 .... we would use a_1 = Using that equation a_12. Be sure to use commas where appropriate! Suppose a math class has 24 students. In how many ways can four students be selected to take part in a survey? In how many ways can four students be selected to do 4 different mathematical problems on the board? Use commas where appropriate.

Solution

1)given terms -19, -14, -9, -4,-1.....

from above we can get common difference d = 5, and first term of the series a₀ = -19

so n^th term of the series a_n = a₀ +(n-1)d

                                         = -19 +(n-1)5

                                         = -19+5n -5

                                          =5n -24

hence a₃₆ = 5*36 -24 =156

and for a Arithematic progression sum to n^th term S_n = n(a₀ +a_n)/2

                                                                             =36(-19+156)/2

                                                                              = 2466

2)series: 3, 12, 48, 192.....

=> 3, 3*4 , 3*4² , 3*4³ ......

so it is a GP with first term a = 3 and r = 4

n^th term of a GP = ar^n-1

so 12^th = 3(4)^12-1 = 3* 4¹¹ = 12582912

out of 24 students 4 students can be selected in ²⁴C₄ ways

and to do four different problem

²⁴C₁ * ²³C₁ * ²²C₁ * ²¹C₁

3)