The 3rd and 8th term of an arithmetic progression A.P are -9 and 26.what their common difference nd first term?

the terms differ by 5d, so 5d=35, d=7

a+2d = a+14 = -9, so a = -23

-23 -16 -9 -2 5 12 19 26 33 ...

The 4th term of an a.p is 132 and 7th term is 260.find the 1st term the common difference and the 37th term.

To find the common difference (d) and the first term (a) of an arithmetic progression (A.P.), we can use the formulas:

n-th term of A.P. (Tn) = a + (n-1)d ----- (1)

where 'n' is the term number.

We are given that the 3rd term (T3) is -9 and the 8th term (T8) is 26. We can use equation (1) to find the values of 'd' and 'a'.

Step 1: Find the common difference (d)
T3 = a + (3-1)d = -9
T8 = a + (8-1)d = 26

Subtracting the first equation from the second equation, we get:
T8 - T3 = (a + (8-1)d) - (a + (3-1)d)
26 - (-9) = a + 7d - (a + 2d)
35 = 5d

Dividing both sides by 5, we get:
d = 7

Step 2: Find the first term (a)
Substituting the value of 'd' (7) in one of the equations, we can solve for 'a'.

T3 = a + (3-1)d = -9

Substituting the values, we get:
-9 = a + 2(7)
-9 = a + 14

Solving the equation, we get:
a = -9 - 14
a = -23

Therefore, the common difference (d) is 7 and the first term (a) is -23 in the given arithmetic progression.

To find the common difference and the first term of an arithmetic progression (A.P.), we can use the formula for the nth term of an A.P., which is:

aₙ = a₁ + (n-1)d

Where:
aₙ = nth term
a₁ = first term
d = common difference
n = position of the term

Given that the 3rd term and 8th term are -9 and 26 respectively, we can substitute these values into the formula.

For the 3rd term:
a₃ = -9
n = 3

-9 = a₁ + (3-1)d
-9 = a₁ + 2d (Equation 1)

For the 8th term:
a₈ = 26
n = 8

26 = a₁ + (8-1)d
26 = a₁ + 7d (Equation 2)

Now we have a system of two equations with two variables (a₁ and d). We can solve this system of equations using substitution or elimination.

Let's use the substitution method and solve for a₁ in Equation 1:

-9 = a₁ + 2d
a₁ = -9 - 2d (Equation 3)

Substitute Equation 3 into Equation 2:

26 = (-9 - 2d) + 7d
26 = -9 + 5d
35 = 5d
d = 7

Now we can substitute the value of d back into Equation 3 to find a₁:

a₁ = -9 - 2(7)
a₁ = -9 - 14
a₁ = -23

Therefore, the common difference is 7 and the first term is -23.