The 14th term of an A.P is 96,while 25th term is 173 find the ,19th term,sum of 13th and 56th terms,product of 6th and 13th terms.
Well, if I were an arithmetic progression, I would probably be the "Slowly Running Out of Jokes" series. But don't worry, I'll still try to give you some funny answers!
Let's start with finding the 19th term. We know that the 14th term is 96. Since it's an A.P., we can say that the common difference (the difference between consecutive terms) is the same throughout. So, let's find the common difference first.
The difference between the 14th term and the 25th term is 173 - 96 = 77. Since there are 11 terms between the 14th and 25th term, the common difference is 77 / 11 = 7.
Now, using the common difference, we can find the 19th term:
19th term = 14th term + (19 - 14) * common difference
19th term = 96 + 5 * 7
19th term = 96 + 35
19th term = 131
Next, let's find the sum of the 13th and 56th terms. We know that the common difference is 7, so:
13th term = 14th term - common difference
13th term = 96 - 7
13th term = 89
56th term = 13th term + (56 - 13) * common difference
56th term = 89 + 43 * 7
56th term = 89 + 301
56th term = 390
Sum of the 13th and 56th terms = 89 + 390 = 479
Lastly, let's find the product of the 6th and 13th terms:
6th term = 14th term - (14 - 6) * common difference
6th term = 96 - 8 * 7
6th term = 96 - 56
6th term = 40
Product of the 6th and 13th terms = 40 * 89 = 3560
I hope you find these answers as amusing as I find myself!
To find the common difference (d) of an arithmetic progression (A.P), we can use the formula:
d = (nth term - 1st term) / (n - 1)
where nth term is the given term, and n is the position of that term in the series.
Let's start by finding the common difference (d) using the given 14th term and 25th term:
d = (25th term - 1st term) / (25 - 1)
d = (173 - 96) / 24
d = 77 / 24
To find the 19th term of the A.P, we can use the formula:
nth term = 1st term + (n - 1) * d
where nth term is the term we want to find, n is the position of that term, and d is the common difference.
19th term = 1st term + (19 - 1) * d
19th term = 96 + 18 * (77 / 24)
To find the sum of the 13th and 56th terms, we can use the formula:
sum = (n/2) * (first term + last term)
where n is the total number of terms, first term is the initial term, and last term is the final term.
sum = (56/2) * (96 + (56 - 1) * d)
To find the product of the 6th and 13th terms, we can simply multiply the values:
product = 6th term * 13th term
Now, plugging in the values from the calculations we've just done, we can find the answers to your questions.
To find the 19th term of the arithmetic progression (AP), we can use the formula:
an = a + (n - 1)d,
where an is the nth term, a is the first term, n is the term number, and d is the common difference.
1. Let's start by finding the common difference (d):
To find the common difference, we can subtract the 14th term from the 25th term:
d = 173 - 96 = 77.
2. Now, we can find the first term (a) using the newly found common difference:
a = 96 - (14 - 1) * 77 = 96 - 13 * 77 = 96 - 1001 = -905.
3. To find the 19th term, we can now substitute the values into the formula:
a19 = -905 + (19 - 1) * 77 = -905 + 18 * 77 = -905 + 1386 = 481.
Therefore, the 19th term of the AP is 481.
Next, let's find the sum of the 13th and 56th terms.
4. We can use the formula for the nth term again to find the 56th term (a56):
a56 = -905 + (56 - 1) * 77 = -905 + 55 * 77 = -905 + 4235 = 3325.
5. Now, we can find the sum of the 13th and 56th terms:
sum = a13 + a56 = -905 + 12 * 77 + 3325 = -905 + 924 + 3325 = 4344.
Therefore, the sum of the 13th and 56th terms is 4344.
Lastly, let's find the product of the 6th and 13th terms.
6. To find the 6th term (a6), we substitute n = 6 into the nth term formula:
a6 = -905 + (6 - 1) * 77 = -905 + 5 * 77 = -905 + 385 = -520.
7. To find the product of the 6th and 13th terms:
product = a6 * a13 = -520 * (-905) = 471,400.
Therefore, the product of the 6th and 13th terms is 471,400.
To summarize:
- 19th term: 481
- Sum of 13th and 56th terms: 4344
- Product of 6th and 13th terms: 471,400.