THE SUM OF THE 2ND AND 5TH TERMS OF AN A.P IS 42.

IF THE DIFFERENCE BETWEEN THE 6TH AND 3RD TERM IS 12,
FIND THE:
COMMON DIFFERENCE,
1ST TERM,
20TH TERM.

Explain in a simpler way please

I didn't understand

Illustrate

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Understood

Thanks alot

Please break it down to ss1 level of understanding

Let

d = the common difference
t(n)=nth term

t(6)-t(5)=d
t(5)-t(4)=d
t(4)-t(3)=d
Add
t(6)-t(3)=3d
or
3d=12
Solve for the common difference d=4

Sum of the 2nd and 5th terms = 42
t(2)+t(5)
=t(2)+(t(2)+3d)
=2*t(2)+12 = 42
Solve for t(2) = 15

t(1) should be t(2)-1=15-4=11
and
t(20)=t(1)+19d

After the step where you expande t5 i just got lost

please break it down into a more simple form.

To find the common difference, first, we need to understand the concept of an Arithmetic Progression (A.P). An A.P is a sequence of numbers in which the difference between any two consecutive terms is constant.

Let's denote the common difference by 'd' and the first term by 'a'.

The formula for the n-th term of an A.P is given by:

(1) Tn = a + (n-1)d

where:
Tn = n-th term
a = first term
d = common difference
n = position of the term

Now, let's solve the given question step by step.

1. The sum of the 2nd and 5th terms of the A.P is 42.

The sum of the 2nd and 5th terms can be expressed as follows:

(2) (a + d) + (a + 4d) = 42

Simplifying equation (2), we get:

(3) 2a + 5d = 42

2. The difference between the 6th and 3rd terms of the A.P is 12.

The 6th term of the A.P can be expressed as:

(4) a + 5d

The 3rd term of the A.P can be expressed as:

(5) a + 2d

The difference between the 6th and 3rd terms can be expressed as follows:

(6) (a + 5d) - (a + 2d) = 12

Simplifying equation (6), we get:

(7) 3d = 12

Solving equation (7), we find that d = 4.

Now, substituting the value of d = 4 into equation (3), we can solve for a:

2a + 5d = 42
2a + 5(4) = 42
2a + 20 = 42
2a = 42 - 20
2a = 22
a = 11

Therefore, the common difference is 4, the first term is 11, and we need to find the 20th term.

To find the 20th term (T20), we can use equation (1):

T20 = a + (n-1)d
T20 = 11 + (20-1)(4)
T20 = 11 + 19(4)
T20 = 11 + 76
T20 = 87

Therefore, the 20th term of the A.P is 87.