If the 5th term of an arithmetic progression is 20 and the 10th term is 35, what

If the 5th term of an arithmetic progression is 20 and the 10th term is 35, what is the first term?

If the 5th term of an arithmetic progression is 20 and the 10th term is 35, what is the first term?

Step 1: Understand that in an arithmetic progression (AP), each term can be expressed using the first term (a) and the common difference (d).
Step 2: The formula for the nth term of an AP is given by: nth term = a + (n-1)d.
Step 3: For the 5th term, substitute n = 5 into the formula: 5th term = a + (5-1)d = a + 4d.
Step 4: We know from the question that the 5th term is 20, so we can write the equation: a + 4d = 20.
Step 5: For the 10th term, substitute n = 10 into the formula: 10th term = a + (10-1)d = a + 9d.
Step 6: We know from the question that the 10th term is 35, so we can write the equation: a + 9d = 35.
Step 7: Now we have two equations: a + 4d = 20 and a + 9d = 35.
Step 8: To find the common difference (d), subtract the first equation from the second: (a + 9d) - (a + 4d) = 35 - 20.
Step 9: This simplifies to 5d = 15, so d = 3.
Step 10: Now that we have d, substitute it back into one of the original equations to find a. Using a + 4d = 20: a + 4(3) = 20.
Step 11: This simplifies to a + 12 = 20, so a = 20 - 12.
Step 12: Therefore, a = 8. The first term is 8.

Arithmetic Progression (AP) – An arithmetic progression is a sequence of numbers in which the difference between consecutive terms is constant.
Term Calculation in AP – The nth term of an arithmetic progression can be calculated using the formula: T_n = a + (n-1)d, where a is the first term and d is the common difference.
Simultaneous Equations – The problem requires solving a system of linear equations to find the values of a and d.