The first Last term of an arithmetic progression are 7 and 43 respectively. The sum of the terms of the progression is 250 Find the number of terms and common difference of the progression
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Let's solve this problem step-by-step.
Given:
First term (a) = 7
Last term (L) = 43
Sum of the terms (Sₙ) = 250
Step 1: Find the number of terms (n).
The formula for the sum of an arithmetic progression when the first and last terms are known is:
Sₙ = (n/2)(a + L)
Substitute the given values into the formula:
250 = (n/2)(7 + 43)
250 = (n/2)(50)
Multiply both sides by 2:
500 = n × 50
Divide by 50 to find n:
n = 500 / 50
n = 10
Step 2: Find the common difference (d).
The formula for the last term of an arithmetic progression is:
L = a + (n - 1)d
Substitute the known values (a, L, and the n we just found) into the formula:
43 = 7 + (10 - 1)d
43 = 7 + (9)d
Subtract 7 from both sides:
43 - 7 = 9d
36 = 9d
Divide by 9 to find d:
d = 36 / 9
d = 4
The number of terms is 10 and the common difference is 4.
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