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The n term of a geometric sequence is denoted by Tn and the sum of the first n terms is denoted by Sn.Given T6-T4=5/2 and S5-S3=5.Calculate (a)the common ratio. (b)the first term of this geometric sequence

The n term of a geometric sequence is denoted by Tn and the sum of the first n terms-example-1

1 Answer

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1 step:
S_(5)=T_(1)+T_(2)+T_(3)+T_(4)+T_(5),
S_(3)=T_(1)+T_(2)+T_(3), then

S_(5)-S_(3)=T_(4)+T_(5)=5.

2 step:
T_(n)=T_(1)*q^(n-1), then

T_(6)=T_(1)*q^(5)

T_(5)=T_(1)*q^(4)

T_(4)=T_(1)*q^(3)

T_(3)=T_(1)*q^(2)
and
\left \{ {{T_(6)-T_(4)= (5)/(2) } \atop {T_(5)+T_(4)=5}} \right. will have form
\left \{ {{T_1*q^(5)-T_(1)*q^(3)= (5)/(2) } \atop {T_(1)*q^(4)+T_(1)*q^(3)=5} \right..

3 step: Solve this system
\left \{ {{T_1*q^(3)*(q^(2)-1)= (5)/(2) } \atop {T_(1)*q^(3)*(q+1)=5} \right. and dividing first equation on second we obtain
(q^(2)-1)/(q+1)= ( (5)/(2) )/(5). So,
((q-1)(q+1))/(q+1) = (1)/(2) and
q-1= (1)/(2),
q= (3)/(2) - the common ratio.

4 step: Insert
q= (3)/(2)into equation
T_(1)*q^(3)*(q+1)=5 and obtain
T_(1)* (27)/(8)*( (3)/(2)+1 ) =5, from where
T_(1)= (16)/(27).




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