Determination of a sequence

A sequence is given whose first three members are a1= 70, a2=64, a3=58.

1, decide whether it is an arithmetic or a geometric sequence and accordingly determine the value of d or q.
2, determine the value of the fifteenth member of the sequence.
3, determine which member of the sequence has the value -50.
4, determine how many members the sequence has in total, if the sum of its members is - 130 and the value of the last member is - 80.

Final Answer:

d =  -6
a₁₅ =  -14
n =  21
n₂ =  26

Step-by-step explanation:

$$\begin{gathered} a_1=70 \\ {a}_2=64 \\ {a}_3=58 \\ {d}_1=a_2-a_1=64-70=-6 \\ {d}_2=a_3-a_2=58-64=-6 \\ {q}_1=a_2/a_1=64/70=\frac{32}{35}\doteq 0.9143 \\ {q}_2=a_3/a_2=58/64=\frac{29}{32}\doteq 0.9063 \\ {q}_1\ne q_2 \\ {d}_1=d_2 \\ d=d_1=(-6)=-6 \end{gathered}$$

$a_{15}=a_1+(15-1)\cdot d=70+(15-1)\cdot (-6)=-14$

$$\begin{gathered} an=a_1+(n-1)\cdot d \\ -50=70+(n-1)\cdot (-6) \\ 6n=126 \\ n=\frac{126}{6}=21 \\ n=21 \end{gathered}$$

$$\begin{gathered} s=-130 \\ b=-80 \\ s=(a_1+b)/2\cdot n_2 \\ {n}_2=2\cdot s/(a_1+b)=2\cdot (-130)/(70+(-80))=26 \end{gathered}$$

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