The sum of the first 16th term of an A. P is 240 and the sum of the next 4 term is 220 find the next term of the A. P

65

Step-by-step explanation:

The sum of the first 16 terms of an arithmetic progression (A.P) is 240

The sum of the next 4 terms is 220

The sum of n terms in an A.P is given by;

= n/2(2a + (n - 1)d)

240 = 8(2a + 15d) ... (i)

460 = 10(2a + 19d) .... (ii)

Simplifying this gives;

2a + 15d = 30 ... (i)

2a + 19d = 46 ... (ii)

Subtracting (i) from (ii) we get;

4d = 16

d (common difference) = 4

and a (first term) = (30 - 60)/ 2 = -15

The sequence upto 21 terms is here:

-15, -11, -7, -3, 1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 51, 55, 59, 61, 65

So the next term (21^st term) is 65.

 a₂₁ = 65

Step-by-step explanation:

The Sum of an Arithmetic Progression is the sum of the first term plus the sum of the last term divided by 2 and multiplied by the number of terms.

Let's find the 16th term (n = 16)

Now let's find the sum of the first 16 terms. This will be Equation 1:

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Repeat what we did above for the next 4 terms (n = 17 to n = 20). This will be Equation 2:

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Now we have a system of equations. Solve using the Elimination Method:

2a₁ + 15d = 30   → -1(2a₁ + 15d = 30)     →    -2a₁ - 15d = -30    

2a₁ + 35d = 110  →  1(2a₁ + 35d = 110)    →     2a₁ + 35d = 110  

                                                                             20d = 80

                                                                                  d = 4

Input d = 4 into one the equations to solve for a₁:

Equation 1: 2a₁ + 15d = 30

                  2a₁ + 15(4) = 30

                    2a₁ + 60 = 30

                             2a₁ = -30

                               a₁ = -15

Given a₁ = -15 and d = 4, we can find the next term (n = 21)

see explanation

step-by-step explanation:

since the sequence is geometric there is a common ratio r between consecutive terms, that is

r = 4 ÷ - 2 = - 2

r = - 8 ÷ 4 = - 2

r = 16 ÷ 8 = - 2

the next term - 13 must be incorrect as it should be - 32 for the sequence to be geometric.

the first 4 terms are therefore geometric with common ratio r = - 2

step-by-step explanation:

12: 00 p.m. to 7: 00 p.m. is a total of 7 hours.

4: 00 p.m. to 7: 00 p.m. is a total of 3 hours.

this means that from 4: 00 p.m to 7: 00 p.m. this represents 3/7 of what was lost from 12: 00 p.m. to 7: 00 p.m.

after knowing this, after doing 11,600 - 6,800 = 4,800

4,800 represents 3/7 of what was lost for 7 hours.

we then divide 4,800 by 3 which gets 1,600.

because we know that 1,600 represents 1/7 of how much water was lost, we just have to multiply by 7, which gets 11,200.   after adding 11,200 to how much water was left at 7: 00 p.m. which was 6,800. 11,200 + 6,800 = 18,000

this means that the pool originally had 18,000 gallons in the pool.

you can also check your work by doing 18,000 - 4(1600), which becomes           1800 - 6400 which gets you 11,600.