1) Given the terms a35 = 1238, a70 = 1483 of an AP. Find the exact value of the term a105 a sequence, and sum of sequence.
2) Given the term a3 = 99 and a5 = 891 of GP. Find the exact value of the term a7 of the sequence and the sum of sequence.
To find the exact value of term a105 in an arithmetic progression (AP) and the sum of the AP, we need to use the formula for the nth term and the formula for the sum of n terms in an AP.
1) Arithmetic Progression (AP):
The formula for the nth term of an AP is given by:
an = a1 + (n - 1)d,
where "an" is the nth term, "a1" is the first term, "n" is the position of the term, and "d" is the common difference.
We are given the values of a35 = 1238 and a70 = 1483.
Using these values, we can find the common difference (d):
a35 = a1 + (35 - 1)d = 1238
a1 + 34d = 1238 ------ Equation 1
a70 = a1 + (70 - 1)d = 1483
a1 + 69d = 1483 ------ Equation 2
To eliminate "a1," subtract Equation 1 from Equation 2:
(69d - 34d) = (1483 - 1238)
35d = 245
d = 7
Now, substitute the value of d back into either of the equations to find "a1":
a1 + 34(7) = 1238
a1 + 238 = 1238
a1 = 1238 - 238
a1 = 1000
Now that we have the values of a1 and d, we can find the nth term (a105):
a105 = a1 + (105 - 1)d
a105 = 1000 + (104)(7)
a105 = 1000 + 728
a105 = 1728
Therefore, the exact value of term a105 in the arithmetic progression is 1728.
To find the sum of the sequence:
The formula for the sum of n terms in an AP is given by:
Sn = (n/2)(2a1 + (n - 1)d),
where "Sn" is the sum of the first "n" terms.
Using the given values, we can find the sum of the sequence:
n = 105, a1 = 1000, and d = 7
S105 = (105/2)(2(1000) + (105 - 1)(7))
S105 = (52.5)(2000 + 104(7))
S105 = (52.5)(2000 + 728)
S105 = (52.5)(2728)
S105 = 143040
Therefore, the sum of the sequence is 143040.
2) Geometric Progression (GP):
To find the exact value of term a7 in a geometric progression (GP) and the sum of the GP, we need to use the formula for the nth term and the formula for the sum of n terms in a GP.
The formula for the nth term of a GP is given by:
an = a1 * r^(n - 1),
where "an" is the nth term, "a1" is the first term, "n" is the position of the term, and "r" is the common ratio.
We are given the values of a3 = 99 and a5 = 891.
Using these values, we can find the common ratio (r):
a3 = a1 * r^(3 - 1) = 99
a1 * r^2 = 99 ------ Equation 1
a5 = a1 * r^(5 - 1) = 891
a1 * r^4 = 891 ------ Equation 2
To eliminate "a1," divide Equation 2 by Equation 1:
(a1 * r^4) / (a1 * r^2) = 891 / 99
r^2 = 9
Taking the square root of both sides, we can find the value of r:
r = sqrt(9)
r = 3
Now that we have the value of r, we can find "a1" by substituting it into Equation 1:
a1 * (3^2) = 99
a1 * 9 = 99
a1 = 11
Now, we can find the nth term a7:
a7 = a1 * r^(7 - 1)
a7 = 11 * 3^6
a7 = 11 * 729
a7 = 8019
Therefore, the exact value of term a7 in the geometric progression is 8019.
To find the sum of the sequence:
The formula for the sum of n terms in a GP is given by:
Sn = a1 * (1 - r^n) / (1 - r),
where "Sn" is the sum of the first "n" terms.
Using the given values, we can find the sum of the sequence:
n = 7, a1 = 11, and r = 3
S7 = 11 * (1 - 3^7) / (1 - 3)
S7 = 11 * (1 - 2187) / (1 - 3)
S7 = 11 * (-2186) / (-2)
S7 = (-24046) / (-2)
S7 = 12023
Therefore, the sum of the sequence is 12023.