Given that the first term of an AP is 7 and its 10th term is twice the second term, calculate
a. 19th term b. Sum of the 28th term c. Difference between the 9th and 6th term?
a = 7
a+9d = 2(a+d)
7 + 9d = 14 + 2d
7d = 7
d = 1
Now you have a and d, use your formulas to find all the questions that were asked.
To solve this problem, we can use the formula for the nth term of an arithmetic progression (AP):
a_n = a + (n-1)d
where a_n is the nth term, a is the first term, n is the position of the term, and d is the common difference.
a. To find the 19th term:
We are given that the first term, a, is 7.
Using the given information that the 10th term is twice the second term, we can write:
a + 9d = 2(a + d)
Substituting the value of the first term, we have:
7 + 9d = 2(7 + d)
Simplifying the equation, we get:
7 + 9d = 14 + 2d
7d = 7
d = 1
Now, we can find the 19th term using the formula:
a_n = a + (n-1)d
a_19 = 7 + (19-1)(1)
a_19 = 7 + 18
a_19 = 25
Therefore, the 19th term is 25.
b. To find the sum of the 28th term:
We know that the formula for the sum of the first n terms of an AP is:
S_n = (n/2)(2a + (n-1)d)
We are given that the first term, a, is 7, and we have already calculated that the common difference, d, is 1.
Now we can substitute these values into the formula and calculate the sum of the 28th term:
S_28 = (28/2)(2(7) + (28-1)(1))
S_28 = 14(14 + 27)
S_28 = 14(41)
S_28 = 574
Therefore, the sum of the 28th term is 574.
c. To find the difference between the 9th and 6th term:
We can use the formula for the nth term to calculate the 9th and 6th terms:
a_9 = a + (9-1)d
a_6 = a + (6-1)d
Using the given information, we can set up the equation:
a + 8d = 2(a + d)
Simplifying the equation, we get:
a + 8d = 2a + 2d
6d = a
Now we can substitute this value into the equation for the difference between the 9th and 6th terms:
a_9 - a_6 = (a + 8d) - (a + 5d)
= a - a + 8d - 5d
= 3d
Substituting the value of the common difference, we have:
a_9 - a_6 = 3(1)
= 3
Therefore, the difference between the 9th and 6th term is 3.