Sequence sum formula

Two arithmetic sequences have the same first term. The nth term of the first sequence is 15, and of the second sequence, 21. The sum of the first n terms of the first sequence is 63, and of the second sequence, 84. Write the sums of the first n terms of both sequences

Final Answer:

a₁ =  3
b₁ =  3

Step-by-step explanation:

$$\begin{gathered} a_1=b_1 \\ a=15 \\ b=21 \\ {s}_1=63 \\ {s}_2=84 \\ {s}_1=(a_1+a)/2\cdot n \\ {s}_2=(a_1+b)/2\cdot n \\ {s}_1/s_2=(a_1+a)/(a_1+b) \\ {s}_1\cdot (a_1+b)=(a_1+a)\cdot s_2 \\ 63\cdot (a_1+21)=(a_1+15)\cdot 84 \\ 21a_1=63 \\ {a}_1=\frac{63}{21}=3 \\ {a}_1=3 \\ {s}_1\cdot a_1+s_1\cdot b=a_1\cdot s_2+a\cdot s_2 \\ {a}_1=\frac{a\cdot s_2-s_1\cdot b}{s_1-s_2}=\frac{15\cdot 84-63\cdot 21}{63-84}=3 \\ {a}_1=n=2\cdot s_1/(a_1+a)=2\cdot 63/(3+15)=7=3 \end{gathered}$$

$b_1=a_1=3$

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