Given that 1st term of an A.P is 7 and it's 10th term is twice the second term,calculate the (a) 19th term (b)sum of 28 terms (c) difference between 9th and 6th terms

1st term=7 2nd term=a+d

10term=a+9d
7+9d=2(14+2d)
14-7=9d-2d
7=7d=d=1
sum of 28 term
n/2(2a+(n-1)d)
28/2(2*7+(28-1)1)
14(14+27)=14(41)
25term=574
9th term=7+(9-1)1
7+(9-1)
7+8=15
6th term=7+(6-1)1
7+(6-1)=7+5=12
9th term-6th term=15-12=3.

Statistics??

What's the SUBJECT??

Oh, I see we're playing with numbers! How delightful! Let's get started with these calculations.

(a) To find the 19th term, we need to determine the common difference first. We know that the 10th term is twice the second term, so we can say:

10th term = 2nd term

Now, let's find the difference between the 10th and 1st terms:

10th term - 1st term = (2nd term - 1st term)

Using the given information:

2nd term - 1st term = 10th term - 1st term

Using the formula for the nth term of an arithmetic progression:

9d = 0

Therefore, d = 0.

Since the common difference is 0, each term of the sequence is the same. Hence, the 19th term will also be 7.

(b) For the sum of 28 terms, we can use the formula:

Sum = (n/2)(2a + (n-1)d)

Given a = 7 and n = 28, we already determined that d = 0. Let's input these values:

Sum = (28/2)(2(7) + (28-1)(0))
Sum = 14(14) = 196

The sum of 28 terms is 196.

(c) To find the difference between the 9th and 6th terms, let's consider their values. The 9th term will be equal to the first term because the common difference is 0. Hence, we have:

9th term = 7

The 6th term will be calculated using the formula for the nth term:

6th term = 1st term + (n - 1)d
6th term = 7 + (6 - 1)(0)
6th term = 7

Therefore, the difference between the 9th and 6th terms is 0.

I hope my clownish calculations brought a smile to your face!

To find the solution to this problem, we can use the formulas for the nth term and the sum of an arithmetic progression (A.P.). The nth term formula is given by:

an = a + (n - 1)d

Where:
an = nth term
a = first term
n = position of the term
d = common difference

In this case, we are given that the first term (a) is 7, and the 10th term is twice the second term. Let's denote the second term as a2.

Given:
a = 7
a10 = 2a2

We can find the common difference (d) by subtracting the first term from the second term:

a2 = a + (2 - 1)d
2 = 7 + d
d = 2 - 7
d = -5

Now that we have the common difference, we can find the 19th term (a19) using the formula:

a19 = a + (19 - 1)d

(a) To find the 19th term:
a19 = 7 + (19 - 1)(-5)
a19 = 7 + 18(-5)
a19 = 7 - 90
a19 = -83

Therefore, the 19th term is -83.

To find the sum of the first 28 terms, we can use the sum formula for an A.P.:

Sn = (n/2)(2a + (n - 1)d)

(b) To find the sum of the first 28 terms:
S28 = (28/2)(2(7) + (28 - 1)(-5))
S28 = 14(14 - 135)
S28 = 14(-121)
S28 = -1,694

Therefore, the sum of the first 28 terms is -1,694.

To find the difference between the 9th term (a9) and the 6th term (a6), we can use the nth term formula:

(a) To find the difference between the 9th and 6th terms:
a9 = a + (9 - 1)d
a6 = a + (6 - 1)d

Substituting the given values:
a9 = 7 + 8(-5)
a6 = 7 + 5(-5)

a9 = 7 - 40
a6 = 7 - 25

We can now calculate the difference:
a9 - a6 = (-33) - (-18)
a9 - a6 = -33 + 18
a9 - a6 = -15

Therefore, the difference between the 9th and 6th terms is -15.

a10=a1+r(10-1)

14=7+r(9)
r=7/9
check
7, 7 7/9, 7 14/9, 7 21/9, 7 28/9, 7 35/9, 7 42/9, 7 49/9, 7 56/9, 14

Use the formulas for sum, knowing r, a1