First 5 terms is sum 100 and 10 terms sum is 150 find the arithmetic sequence
Answers
Step-by-step explanation:
How do you calculate this? The sum of the first 5 terms of an arithmetic series is 110 and the sum of the first 10 terms is 320. How do I go about finding the first term and the common difference?
In an arithmetic sequence the terms are; a, a+d, a+2d, a+3d, … and so on.
The sum of the first 5 terms would be: n + (n+d) + (n+2d) + (n+3d) + (n+4d), which is equal to 5n + 10.
The sum of the first 10 terms would be: n + (n+d) + (n+2d) + (n+3d) + (n+4d) + (n+5d) + (n+6d) + (n+7d) + (n+8d) + (n+9d), which is equal to 10n + 45.
This means you now have two equations and two unknowns, solve this system of equations:
5n + 10d = 110 : Multiply this equation by -2 and add it to the 2nd equation
10n + 45d = 320
-10n - 20d = -220
10n + 45d= 320
25d = 100 => divide both sides by 25
d = 4: The common difference is 4
Substitute 4 in as d to find n.
5n + 10(4) = 110
5n + 40 = 110 : Subtract 40 from both sides
5n = 70 : Divide both sides by 5
n = 14 : The first term is 14
Check
14 + 18 + 22 + 26 + 30 = 110 : checks out
14 + 18 + 22 + 26 + 30 + 34 + 38 + 42 + 46 + 50 = 320 : checks out
The first term is 14, the common difference is 4.