arithmetic sequences

Question

in this arithmetic sequence t₄ = t₁² if each of its terms is decreased by 6 then its seventh term will be equal to three times as much as its fourth term . write down the first 5 terms then find the sum of the first 10 terms of this sequence.

Answer 1

Hassan,

 

t4 = t1 + 3*d

 

Since t4 = t1^2 we have t1^2 = t1 + 3*d as the first equation

 

Also, t7 = t4 + 3*d

 

And t7-6 = 3*(t4-6)

 

Substitute for t7 from previous equstion

 

t4 + 3*d - 6 = 3t4 - 18

 

2t4 -3*d -12 = 0

 

Substitute for t4 in the equation

2*(t1 + 3*d) - 3*d -12 = 0

2t1 + 3d -12 = 0

 

3d = 12 - 2*t1

 

Replace 3d from eq 1

 

t1^2 = t1 + 12 - 2t1

 

t1^2 + t1 -12 = 0

 

(t1+4)(t1-3) = 0

 

t1 = -4, or t1 = 3

 

d = 1/3(12-2*t1) = 20/3 or 2

 

So t1 = 3 and d = 2, hence the first 5 terms are 3, 5, 7, 9 and 11

 

Now you can find the sum of first 10 terms