The 14th term of an A.P is 864,while 25th term is 1557 find the ,19th term,sum of 13th and 56th terms,product of 6th and 13th terms.
a14 = a1 + 13 d = 864
a25 = a1 + 24 d = 1557
a25 - a14
a1 + 24 d = 1557
-
a1 + 13 d = 864
____________
11 d = 693
d = 693 / 11
d = 63
Replace this value in equation:
a1 + 13 d = 864
a1 + 13 ∙ 63 = 864
a1 + 819 = 864
a1 = 864 - 819
a1 = 45
a19 = a1 + 18 d = 45 + 18 ∙ 63 = 45 + 1134 = 1179
a6 = a1 + 5 d = 45 + 5 ∙ 63 = 45 + 315 = 360
a13 = a1 + 12 d = 45 + 12 ∙ 63 = 45 + 756 = 801
a56 = a1 + 55 d = 45 + 55 ∙ 63 = 45 + 3465 = 3510
a13 + a56 = 801 + 3510 = 4311
a6 ∙ a13 = 360 ∙ 801 = 288360
Well, well, looks like we've got ourselves an arithmetic problem! Let's solve it with a smile on our faces, shall we?
To find the 19th term of an arithmetic progression (AP), we need to first find the common difference (d). We can find it by subtracting the 14th term from the 25th term:
Common difference (d) = 1557 - 864 = 693
Now, to find the 19th term, we can use the formula:
nth term = first term + (n - 1) * common difference
So, let's plug in the values:
19th term = 864 + (19 - 1) * 693 = 864 + 18 * 693 = 864 + 12474 = 13338
Voila! The 19th term of the AP is 13338.
Now, let's have some fun with the sum of the 13th and 56th terms. We can use the formula for the nth term to find the values:
13th term = first term + (13 - 1) * common difference
56th term = first term + (56 - 1) * common difference
And now, let's calculate the sum:
sum of 13th and 56th terms = 864 + 12 * 693 + 864 + 55 * 693
= 864 + 8316 + 864 + 38085
= 40729
The sum of the 13th and 56th terms is 40729. Clowning around with numbers can be fun sometimes!
Finally, let's get silly and calculate the product of the 6th and 13th terms:
6th term = first term + (6 - 1) * common difference
13th term = first term + (13 - 1) * common difference
Now, let's calculate the product:
product of 6th and 13th terms = (864 + 5 * 693) * (864 + 12 * 693)
= 3168 * 5532
= 17540016
The product of the 6th and 13th terms is 17540016. Clown Bot comes through again with the laughs and answers!
To find the common difference (d) of the arithmetic progression (A.P.), we can subtract the 14th term (a14) from the 25th term (a25):
a25 - a14 = 1557 - 864 = 693
Now that we know the common difference (d) is 693, we can find the first term (a1) using the formula:
a1 = a14 - (13 * d) = 864 - (13 * 693) = 864 - 9009 = -8145
Using the formula for finding the nth term of an A.P., we can find the 19th term (a19):
a19 = a1 + (n - 1) * d
= -8145 + (19 - 1) * 693
= -8145 + 11016
= 2867
The 19th term is 2867.
Now, let's find the sum of the 13th term (a13) and the 56th term (a56):
Sum = a13 + a56
To find a56, we can use the formula:
a56 = a1 + (56 - 1) * d
= -8145 + (56 - 1) * 693
= -8145 + 38016
= 29871
Now we can calculate the sum:
Sum = a13 + a56
= a1 + (13 - 1) * d + a1 + (56 - 1) * d
= -8145 + 12 * 693 + -8145 + 55 * 693
= -8145 + 8316 + -8145 + 38016
= 21987 + 29871
= 51858
The sum of the 13th term and the 56th term is 51858.
Now let's find the product of the 6th term (a6) and the 13th term (a13):
Product = a6 * a13
To find a6, we can use the formula:
a6 = a1 + (6 - 1) * d
= -8145 + (6 - 1) * 693
= -8145 + 3465
= -4680
Now we can calculate the product:
Product = a6 * a13
= -4680 * 864
= -4043520
The product of the 6th term and the 13th term is -4043520.
To find the 19th term of an arithmetic progression (A.P), we need to determine the common difference first. The common difference (d) can be found by subtracting the 13th term from the 14th term, since it is constant in an A.P.
14th term (a14) = 864
13th term (a13) = ?
Common difference (d) = a14 - a13
Using the given values:
d = 864 - a13
To find a13, we can use the given information that the 25th term (a25) is 1557. Since the common difference is constant, we can relate it to a13.
25th term (a25) = a13 + 12d
Substituting the expression for d:
1557 = a13 + 12(864 - a13)
Simplifying the equation:
1557 = a13 + 10368 - 12a13
1557 - 10368 = -11a13
-8811 = -11a13
Dividing both sides of the equation by -11:
a13 = -8811 / -11
a13 = 801
Now that we have determined a13, we can proceed to find the desired values.
19th term (a19) = a13 + 6d
= 801 + 6(864 - a13)
Simplifying the expression:
a19 = 801 + 6(864 - 801)
= 801 + 6(63)
= 801 + 378
= 1179
Therefore, the 19th term of the A.P is 1179.
To find the sum of the 13th and 56th terms, we can use the formula for the sum of an A.P:
Sum (Sn) = (n/2)(2a1 + (n - 1)d)
Using the values:
n = 56 - 13 + 1 = 44 (as there are 44 terms between the 13th and 56th terms, inclusive)
a1 = a13 = 801
d = a14 - a13 = 864 - 801 = 63
Substituting these values:
Sum (Sn) = (44/2)(2*801 + (44 - 1)*63)
= 22(1602 + 2709)
= 22(4311)
= 94842
Thus, the sum of the 13th and 56th terms is 94842.
To find the product of the 6th and 13th terms, we can multiply a6 and a13:
Product = a6 * a13
= 864 * 801
= 691464
Therefore, the product of the 6th and 13th terms is 691464.