Answer:
The first term is 3 and the common difference is 2
Step-by-step explanation:
The formula of the sum of the nth term of the arithmetic sequence is
[tex]S_{n}=\frac{n}{2}[2a+(n-1)d][/tex], where
∵ The sum of the first five terms of an arithmetic sequence is 35
∴ [tex]S_{5}[/tex] = 35
→ Substitute The values of n and S in the rule above
∵ 35 = [tex]\frac{5}{2}[2a+(5-1)d][/tex]
∴ 35 = [tex]\frac{5}{2}[2a+4d][/tex]
→ Simplify the right side
∴ 35 = 5a + 10d
→ Switch the two sides
∴ 5a + 10d = 35 ⇒ (1)
∵ The sum of the next five terms is 85
→ To find the sum of the first 10 terms add 85 to [tex]S_{5}[/tex]
∴ [tex]S_{10}[/tex] = 35 + 85 = 120
→ Substitute The values of n and S in the rule above
∵ 120 = [tex]\frac{10}{2}[2a+(10-1)d][/tex]
∴ 120 = 5[2a + 9d]
→ Simplify the right side
∴ 120 = 10a + 45d
→ Switch the two sides
∴ 10a + 45d = 120 ⇒ (2)
Now we have a system of equations to solve it and find a and d
→ Multiply equation (1) by -2
∵ -2(5a) + -2(10d) = -2(35)
∴ -10a + -20d = -70 ⇒ (3)
→ Add equations (2) and (3)
∵ (10a + -10a) + (45d + -20d) = (120 + -70)
∴ 25d = 50
→ Divide both sides by 25 to find d
∴ d = 2
→ To find a substitute d in equation (1) by 2
∵ 5a + 10(2) = 35
∴ 5a + 20 = 35
→ Subtract 20 from both sides
∵ 5a + 20 - 20 = 35 - 20
∴ 5a = 15
→ Divide both sides by 5
∴ a = 3
∴ The first term is 3 and the common difference is 2
