Solve Q12a & 12c Showing detailly all steps 10. (V Find n, given that (5 – 2r) = -165.* (ii) The sum of the first n terms of a series is 9(1 -3na) Find the nth term of the seriesb) Show that the series is a geometric series and find its sum to infinity11. Given that x – 3, 2x + 1 and 7x – 2 are the first three consecutive terms of an arithmeticprogression, find the value of x and hence obtain the sum of the first ten terms of theprogression.X-3=) converges.12. a) Find the range of values of x for which the series =1X-1X-31c) Determine the value of x for which 1X-1r3D13x3. Find the set of values of x for which the series E1() is convergentr=12+x.V14. The sum of n terms of a certain series is given by 17n - 3n". Find an expression for the nthterm of the series and show that the series is an arithmetic progression.15. Find the set of real values of x for which the geometric seriesMo X# - 2, is convergent.r+2,A6. The sum of the first, third and seventh terms of an arithmetic progression is 25. If thethirteenth term of this progression is three times the fourth term, find the first term and theCommon difference of the progression.The sum of the first n terms of a sequence, S, is given by S, =-(3n + 1). Find the sum of the terms2from the 10" to the 30" term.(b) A geometric progression has its first term and common ratio as sir Land cos 20 respectively, where|cos 20

Question

Solve Q12a & 12c Showing detailly all steps 10. (V Find n, given that (5 – 2r) = -165.* (ii) The sum of the first n terms of a series is 9(1 -3na) Find the nth term of the seriesb) Show that the series is a geometric series and find its sum to infinity11. Given that x – 3, 2x + 1 and 7x – 2 are the first three consecutive terms of an arithmeticprogression, find the value of x and hence obtain the sum of the first ten terms of theprogression.X-3=) converges.12. a) Find the range of values of x for which the series =1X-1X-31c) Determine the value of x for which 1X-1r3D13x3. Find the set of values of x for which the series E1() is convergentr=12+x.V14. The sum of n terms of a certain series is given by 17n - 3n&#34;. Find an expression for the nthterm of the series and show that the series is an arithmetic progression.15. Find the set of real values of x for which the geometric seriesMo X# - 2, is convergent.r+2,A6. The sum of the first, third and seventh terms of an arithmetic progression is 25. If thethirteenth term of this progression is three times the fourth term, find the first term and theCommon difference of the progression.The sum of the first n terms of a sequence, S, is given by S, =-(3n + 1). Find the sum of the terms2from the 10&#34; to the 30&#34; term.(b) A geometric progression has its first term and common ratio as sir Land cos 20 respectively, where|cos 20 < 1. Prove that its sum to infinity is cot 0(

Accepted Answer

12(a)
Given infinite series is:
∑r=1∞x-3x-1r=x-3x-1+x-3x-12+x-3x-13+......
Let is consider,…

12. a) Find the range of values of x for which the series E1 (=) converges. c) Determine the value of x for which E1 ) =