Question

the sum of the first 9 terms of an A.p is 72 and the sum of the next 4 terms is 71. find the A.p. solution

Answer 1

9/2 (2a+8d) = 72

13/2 (2a+12d) = 72+71

Answer 2

Answer that please

Answer 3

[AI-generated answer — Explain Bot]

To find the arithmetic progression (A.P.), we need to determine the common difference (d) and the first term (a₁).

Let's denote the sum of the first 9 terms of the A.P. as S₁ and the sum of the next 4 terms as S₂.

Given that S₁ = 72, we can use the formula for the sum of an A.P.:

S₁ = (n/2)(2a₁ + (n-1)d)

Where n is the number of terms in the sum, a₁ is the first term, and d is the common difference.

We know that n = 9, so we can substitute this into the formula:

72 = (9/2)(2a₁ + (9-1)d)

Simplifying further:

144 = 18(2a₁ + 8d)
8 = 2a₁ + 8d --- (Equation 1)

Next, we are given that the sum of the next 4 terms, S₂, is 71. Using the same formula:

S₂ = (n/2)(2a₁ + (n-1)d)

Now, n = 4, so we can substitute this value:

71 = (4/2)(2a₁ + (4-1)d)
71 = 2(2a₁ + 3d)
71 = 4a₁ + 6d --- (Equation 2)

We now have a system of two equations with two variables, a₁ and d.

We can solve these equations using either elimination or substitution method.

Subtracting Equation 2 from Equation 1, we get:

8 -71 = 2a₁ + 8d - (4a₁ + 6d)
-63 = -2a₁ + 2d

Dividing the equation by -2, we have:

31.5 = a₁ - d

Now, we can substitute this value of a₁ - d in either Equation 1 or Equation 2 to find the values of a₁ and d.

Let's substitute it into Equation 1:

8 = 2a₁ + 8d

Replacing a₁ - d with 31.5:

8 = 2(31.5 + d) + 8d
8 = 63 + 2d + 8d
-55 = 10d
d = -5.5

Now, substitute the value of d into a₁ - d = 31.5:

a₁ - (-5.5) = 31.5
a₁ + 5.5 = 31.5
a₁ = 31.5 - 5.5
a₁ = 26

Therefore, the first term, a₁, of the A.P. is 26 and the common difference, d, is -5.5.