Answer:
The ratio of the first term to the common difference is 1 : 2
Step-by-step explanation:
The rule of the sum of the AP is
[tex]S_{n}=\frac{n}{2}[2a+(n-1)d][/tex] , where
a is the first number
d is the common difference
n is the position of the number
∵ [tex]S_{10}[/tex] = 4 × [tex]S_{5}[/tex] ⇒ (1)
→ Find [tex]S_{10}[/tex] and [tex]S_{5}[/tex]
∵ In [tex]S_{10}[/tex] n = 10
∴ [tex]S_{10}=\frac{10}{2}[2a+(10-1)d][/tex]
∴ [tex]S_{10}=5[2a+9d][/tex]
∴ [tex]S_{10}[/tex] = 10a + 45d ⇒ (2)
∵ In [tex]S_{5}[/tex] n = 5
∴ [tex]S_{5}=\frac{5}{2}[2a+(5-1)d][/tex]
∴ [tex]S_{5}=2.5[2a+4d][/tex]
∴ [tex]S_{5}[/tex] = 5a + 10d ⇒ (3)
→ Substitute (2) and (3) in (1)
∵ 10a + 45d = 4[5a + 10d]
∴ 10a +45d = 20a + 40d
→ Subtract 40d from both sides
∴ 10a + 45d - 40d = 20a + 40d - 40d
∴ 10a + 5d = 20a
→ Subtract 10a from both sides
∴ 10a - 10a + 5d = 20a - 10a
∴ 5d = 10a
→ Divide both sides by 5
∴ d = 2a
→ That means d is twice a or a is half d
∴ a : d = 1 : 2
The ratio of the first term to the common difference is 1 : 2
