1.) The first and last term of an a.p are 8 and 110^3/5
A.) The number of terms in A.P
B.) The sum of all terms
2.) The first and the third term of a G.P are 2 and 2/9 respectively find
1.) The common ratio
2.) The fifth term
*.) Find the sum of the first 15 term in the sequence below -1,1^2/3,4^1/3,7..…
*.) The house on one side of a particular street are assigned odd numbers starting from 11. if the sum of the numbers is 551 how many houses are there
*.) If 6,and 14 are consecutive terms in an A.p find the value of P
*.) Evaluating f(1/2*g(3)/f(4)+g(5).
1.) For an arithmetic progression (A.P), the formula to find the nth term is given by:
an = a1 + (n-1)d
where an is the nth term, a1 is the first term, and d is the common difference.
In this case, a1 = 8 and an = 110^3/5. We can substitute these values into the formula:
110^3/5 = 8 + (n-1)d
To simplify, we can convert 110^3/5 to a decimal:
110^3/5 ≈ 460.58
Substituting this value back into the equation:
460.58 = 8 + (n-1)d
To find the common difference, we need another equation. Let's consider the second term:
a2 = a1 + d
a2 = 8 + d
Since we're given that the common difference is the same for all terms, we can equate a2 and an to find d:
8 + d = 460.58
d ≈ 452.58
Now we can substitute the value of d back into the first equation and solve for n:
460.58 = 8 + (n-1)(452.58)
452.58(n-1) = 452.58
n - 1 = 1
n = 2
A.) The number of terms in the A.P is 2.
To find the sum of all terms, we can use the formula:
Sn = n/2(a1 + an)
where Sn is the sum of the first n terms.
Substituting the known values:
S2 = 2/2(8 + 460.58)
S2 = 1/2(468.58)
S2 = 234.29
B.) The sum of all terms in the A.P is approximately 234.29.
2.) For a geometric progression (G.P), the formula to find the nth term is given by:
an = a1(r^(n-1))
where an is the nth term, a1 is the first term, and r is the common ratio.
In this case, a1 = 2 and a3 = 2/9. We can substitute these values into the formula:
2/9 = 2(r^2)
r^2 = (2/9)/(2)
r^2 = 1/9
r = 1/3
1.) The common ratio is 1/3.
To find the fifth term, we can substitute the known values into the formula:
a5 = a1(r^(5-1))
a5 = 2((1/3)^4)
a5 = 2/81
2.) The fifth term of the G.P is 2/81.
*.) To find the sum of the first 15 terms in the sequence -1, 1^(2/3), 4^(1/3), 7, ..., we need to determine the pattern of the sequence. It appears to be an arithmetic sequence, where the common difference is 3. Let's confirm this:
The first term is -1, and the second term can be obtained by adding 3 to -1:
-1 + 3 = 2
Similarly, the third term can be obtained by adding 3 to the second term:
2 + 3 = 5
Therefore, the sequence is an arithmetic sequence with a common difference of 3. Using the formula for the sum of an arithmetic sequence:
Sn = (n/2)(a1 + an)
a1 = -1
an = a1 + (n-1)d = -1 + (15-1)3 = 41
Substituting the known values:
S15 = (15/2)(-1 + 41)
S15 = 15/2 * 40
S15 = 300
The sum of the first 15 terms is 300.
*.) The house numbers assigned to the odd-numbered side of the street start from 11 and follow an arithmetic sequence. The sum of the numbers is given as 551. Let's use the formula for the sum of an arithmetic sequence to find the number of houses:
Sn = (n/2)(a1 + an)
a1 = 11
an = a1 + (n-1)d
We don't know the common difference, d. So we have:
551 = (n/2)(11 + an)
551 = (n/2)(11 + (11 + (n-1)d))
551 = (n/2)(22 + (n-1)d)
551 = (n/2)(22 + 21d)
Since n is the number of terms, it must be an integer. Using trial and error, we can find that n = 11 satisfies the equation:
551 = (11/2)(22 + 21d)
551 = 11(22 + 21d)
551 = 242 + 231d
d ≈ 1
Therefore, the common difference is approximately 1.
To find the number of houses, we substitute the value of d back into the equation for Sn:
551 = (n/2)(22 + 21)
551 = (n/2)(43)
n/2 ≈ 551/43
n ≈ (2 * 551)/43
n ≈ 24.74
Since the number of houses must be an integer, we round down to the nearest whole number.
There are 24 houses on the odd-numbered side of the street.
*.) Given that 6 and 14 are consecutive terms of an arithmetic progression (A.P), we can use the formula for the nth term to find the common difference:
an = a1 + (n-1)d
For the first term a1 = 6:
6 = 6 + (1-1)d
0 = d
Since the common difference, d, is equal to 0, all the terms in the A.P will be equal to 6.
*.) The expression f(1/2*g(3)/f(4)+g(5) can be evaluated if we have the definitions of the functions f and g. Please provide the definitions of f and g for further assistance.