The 25 term of an Arithmetic Progression is 7 1/2 and the sum of the first 23 terms is 98 1/2. Find the
i. 1st term
ii. common difference
iii. sum of the first 30 terms
25th term = 7 1/2
a + 24 d = 15/2
sum of first 23 terms = 98 1/2
(23/2)(2a + 22d) = 197/2
23(2a+22d) = 197
2a+22d = 197/23
a+11d = 197/46
subtract the two equations
13d = 15/2 - 197/46 = 74/23
d = 74/299
sub back into first equation
a + 24(74/299) = 15/2
a = 933/598
so sum of 30 terms
= 15(2(933/598) + 29(74/299) )
= 92340/598
sure was expecting some nicer numbers, better check my arithmetic
To find the first term, common difference, and the sum of the first 30 terms, we can use the formulas for arithmetic progression.
Let's break down the problem step by step:
Step 1: Finding the common difference (d)
The formula to find the nth term of an arithmetic progression is:
tn = a + (n-1)d
Here, we are given the 25th term (tn = 7 1/2) and the sum of the first 23 terms (S23 = 98 1/2).
Substituting the values into the formula:
7 1/2 = a + (25-1)d
7 1/2 = a + 24d ---(Equation 1)
Step 2: Finding the first term (a)
To find the first term (a), we need to substitute the common difference (d) into equation 1 and solve for a.
Using the values from equation 1:
7 1/2 = a + 24d
Step 3: Finding the sum of the first 30 terms (S30)
The formula to find the sum of an arithmetic progression is:
Sn = (n/2)(2a + (n-1)d)
Here, we want to find the sum of the first 30 terms (S30). We already have the first term (a) and common difference (d).
Using the values:
Sn = (30/2)(2a + (30-1)d)
Sn = 15(2a + 29d)
Sn = 30a + 435d ---(Equation 2)
Now, we have two equations with two unknowns, a and d.
Step 4: Solving the equations
We can solve the equations simultaneously to find the values of a and d.
From equation 1:
7 1/2 = a + 24d ---(Equation 1)
From equation 2:
98 1/2 = 30a + 435d ---(Equation 2)
Now, we can solve this system of equations to find the values of a and d.
Once we have the values for a and d, we can find the first term (i) and the sum of the first 30 terms (iii) using the formulas we discussed above.