Question

Find the 10th term of a nonconstant arithmetic sequence whose 1st term is 3 with the 1st, 4th, and 13th term forming a geometric sequence.
I've tried doing it differently everytime, but I always come to a dead end..

Answer 1

Don't answer this anymore.... I got it

Answer 2

In arithmetic sequence :

an = a1 + ( n - 1 ) d

Where:

a1 is the first term of the sequence

d is the common difference

n is the number of the term


a1 = 3

a4 = 3 + ( 3 - 1 ) d = 3 + 2 d

a13 = 3 + ( 13 - 1 ) d = 3 + 12 d



1st, 4th, and 13th term forming a geometric sequence, so :

a1 = 3

a4 = a1 * r = 3 r

a13 = a1 * r ^ 2 = 3 r ^ 2

r is the common ratio


a4 = a4

3 + 2 d = 3 r

a13 = a13

3 + 12 d = 3 r ^ 2


Now you must solve system of two equations :

3 + 2 d = 3 r

3 + 12 d = 3 r ^ 2


3 + 2 d = 3 r Subtract 3 to both sides

3 + 2 d - 3 = 3 r - 3

2 d = 3 r - 3

2 d = 3 ( r - 1 ) Divide both sides by 2

d = 3 ( r - 1 ) / 2


3 + 12 d = 3 r ^ 2

3 + 12 * 3 ( r - 1 ) / 2 = 3 r ^ 2

3 + 36 ( r - 1 ) / 2 = 3 r ^ 2

3 + 18 ( r - 1 ) = 3 r ^ 2 Divide both sides by 3

1 + 6 ( r - 1 ) = r ^ 2

1 + 6 r - 6 = r ^ 2

6 r - 5 = r ^ 2 Subtract r ^ 2 to both sides

6 r - 5 - r ^ 2 = r ^ 2 - r ^ 2

- r ^ 2 + 6 r - 5 = 0 Multiply both sides by - 1

r ^ 2 - 6 r + 5 = 0

Solutions are :

r = 1 and r = 5


For r = 1

d = 3 ( r - 1 ) / 2 = 3 ( 1 - 1 ) / 2 = 3 * 0 / 2 = 0 / 2 = 0


For r = 5

d = 3 ( r - 1 ) / 2 = 3 ( 5 - 1 ) / 2 = 3 * 4 / 2 = 12 / 2 = 6


You have a nonconstant arithmetic sequence so solution r = 1 and d = 0 you must ignore becouse for d = 0 :

an = a1 + ( n - 1 ) d

an = a1 + ( n - 1 ) * 0

an = a1

This mean for d = 0 all members of sequence are equal a1.
That is a constant arithmetic sequence .


So :

r = 5

d = 6


an = a1 + ( n - 1 ) d

a13 = 3 + ( 13 - 1 ) * 6

a13 = 3 + 12 * 6 = 3 + 72 = 75


By the way a geometric sequence of 1st, 4th, and 13th therm :

a1 = 3

a4 = a1 * r = 3 * 5 = 15

a13 = a1 * r ^ 2 = 3 * 5 ^ 2 = 3 * 25 = 75


So a13 = 75

Answer 3

an = a1 + ( n - 1 ) d

a10 = 3 + ( 10 - 1 ) * 6

a10 = 3 + 9 * 6 = 3 + 54 = 57

Answer 4

"a4 = 3 + ( 3 - 1 ) d = 3 + 2 d "

Should be


a4 = 3 + ( 4 - 1 ) d = 3 + 3 d