So yeah, this one’s hard, I’ll have to think about it

Find a formula for

nth term of the arithmetic sequence as afunction of n, where

a5=190

and a10=115

The terms of an arithmetic sequence are given by:

a_n = a_1 + (n – 1)d, where a_n is the nth term, a_1 is the first term, n is the current number of term, and d is the common difference

we are told:

a_5 = a_1 + (5 – 1)d = 190

that is, a_1 + 4d = 190

a_10 = a_1 + (10 – 1)d = 115

that is, a_1 + 9d = 115

So we obtain the simultaneous equations:

a_1 + 4d = 190 ……………….(1)

a_1 + 9d = 115 ………………..(2)

=> 5d = -75 ...............................(3) = (2) - (1)

=> d = -15

But a_1 + 4d = 190

=> a_1 -4*15 = 190

=> a_1 = 190 + 4*15

=> a_1 = 250

So the terms of this arithmetic series are given by:

a_n = 250 + (n – 1)(-15)

=> a_n = 250 + 15 – 15n

=>a_n = 265 – 15n

1.32454545… = 1 + 32/100 + 4/10^3 + 5/10^4 + 4/10^5 + 5/10^6 + …

Third question

Use the concepts of geometric sequences to write 1.3245 (i.e., 1.324545454545…)

as rational number in the form a/b

where

aandbare integers.

………………= 1 + 32/100 + 45/10^4 + 45/10^6 + 45/10^8 + …

………………= 1 + 32/100 + (45/10^4)(1 + 1/10^2 + 1/10^4 + 1/10^6 + …)

………………= 1 + 32/100 + (45/10^4)(1 + (1/10^2) + (1/10^2)^2 + (1/10^2)^3 + …)

………………= 1 + 32/100 + (45/10^4)(1 + (1/100) + (1/100)^2 + (1/100)^3 + …)

But 1 + (1/100) + (1/100)^2 + (1/100)^3 + … is a geometric series with a_1 = 1 and r = 1/100. Thus it’s infinite sum is given by 1/(1 – 1/100) = 100/99.

So 1.32454545… = 1 + 32/100 + (45/10^4)(100/99)

………………….= 1 + 32/100 + (45/10^2)(1/99)

………………….= 1 + 32/100 + 45/9900

………………….= 1457/1100

a_1 = 0

Fourth question

Let a1=0

a2=1 andak+ 1 =ak+ak- 1 fork³ 2. Write the first 8 terms of this sequence.

a_2 = 1

a_3 = a_(2+1) = a_2 + a_1 = 1 + 0 = 1

a_4 = a_(3+1) = a_3 + a_2 = 1 + 1 = 2

a_5 = a_(4+1) = a_4 + a_3 = 1 + 2 = 3

a_6 = a_(5+1) = a_5 + a_4 = 2 + 3 = 5

a_7 = a_(6+1) = a_6 + a_5 = 5 + 3 = 8

a_8 = a_(7+1) = a_7 + a_6 = 8 + 5 = 13

So the first 8 terms are 0, 1, 1, 2, 3, 5, 8, 13

So you get $1000 raise for every year you work, so the total money you get from raises is 30*1000 = 30000

Fifth question

How much money will I have been paid over my

30 year career if my starting salary is $40,000, and I

receive a $1,000 salary raise for each year I work

Now you have 40000 base pay, so after 30 years that becomes 30*40000 = 1200000

So your total pay is 30000 + 1200000 = 1230000

So over your 30 year career you were paid $1230000, not bad. Well, it wouldn’t be bad if you weren’t in this economy