In a GP 72+144

In a geometric progression, the sum of the 2nd and 5th terms is 72, and the sum of the 3rd and 6th terms is 144. Find the common ratio, the first term, and the sum of the first six terms.

Final Answer:

q =  2
a₁ =  4
s₆ =  252

Step-by-step explanation:

$$\begin{gathered} s_{25}=72 \\ {s}_{36}=144 \\ {a}_1\cdot (q+q^4)=s_{25} \\ {a}_1\cdot (q^2+q^5)=a_1\cdot q(q+q^4)=s_{36} \\ q\cdot s_{25}=s_{36} \\ q=s_{36}/s_{25}=144/72=2 \end{gathered}$$

$a_1=\frac{s_{25}}{q+q^4}=\frac{72}{2+2^4}=4$

$s_6=a_1\cdot (1+q+q^2+q^3+q^4+q^5)=4\cdot (1+2+2^2+2^3+2^4+2^5)=252$

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