The 14th term of an AP is 96/while the 25th terms is 173/find the Sum of 13th and 56th term.
Let the first term of the AP be 'a' and the common difference be 'd'.
Then, from the given information, we can form the following equations:
a + 13d = 96 (1)
a + 24d = 173 (2)
Subtracting equation (1) from (2), we get:
11d = 77
d = 7
Substituting this value of 'd' in equation (1), we get:
a + 91 = 96
a = 5
Now, we can find the 13th term and 56th term as follows:
13th term = a + 12d = 5 + 12(7) = 89
56th term = a + 55d = 5 + 55(7) = 390
Therefore, the sum of 13th and 56th term is:
89 + 390 = 479. Answer: \boxed{479}.
To find the sum of the 13th and 56th term of an arithmetic progression (AP), we need to first find the common difference of the given AP.
Let's denote the first term as 'a' and the common difference as 'd'.
We are given that the 14th term of the AP is 96, which can be expressed as:
a + (13 - 1)d = 96 --- (Equation 1)
We are also given that the 25th term of the AP is 173, which can be expressed as:
a + (25 - 1)d = 173 --- (Equation 2)
To solve these equations, we can subtract Equation 1 from Equation 2 to eliminate 'a':
(25 - 1)d - (13 - 1)d = 173 - 96
24d - 12d = 77
12d = 77
Dividing both sides of the equation by 12:
d = 77 /12
d = 6.4167 (rounded to four decimal places)
Now that we know the common difference, we can find the first term 'a' by substituting the value of 'd' into either Equation 1 or 2. We will use Equation 1:
a + (13 - 1)(6.4167) = 96
a + (12)(6.4167) = 96
a + 77 = 96
a = 96 - 77
a = 19
So, the first term 'a' is 19 and the common difference 'd' is 6.4167.
To find the sum of the 13th and 56th term, we can use the formula for the sum of an arithmetic series:
Sn = (n/2)(2a + (n-1)d)
Let's calculate the sum for the 13th and 56th term:
For the 13th term:
n = 13
Sn = (13/2)(2*19 + (13-1)*6.4167)
= (13/2)(38 + 12*6.4167)
= (13/2)(38 + 77.0004)
= (13/2)(115.0004)
= 747.0026
For the 56th term:
n = 56
Sn = (56/2)(2*19 + (56-1)*6.4167)
= (56/2)(38 + 55*6.4167)
= (56/2)(38 + 353.1667)
= (56/2)(391.1667)
= 10965.67
Therefore, the sum of the 13th and 56th term of the given AP is 747.0026 + 10965.67 = 11712.6726.