Given that 6,9,13½,21/4,form a sequence of number. Find the sum of the 13th term
If the pattern is correct, your 4th term should be 20 1/4 or 20.5 , not 21/4
looks like a GS, with a = 6 and r = 1.5
sum(13) = a( r^13 - 1)/(r-1) = 6(( 3/2)^13 - 1)/(3/2 - 1)
= 6(1594323/8192 - 1)/ (1/2)
= .....
Yes
Well, to find the sum of the 13th term of the sequence 6, 9, 13½, 21/4, we could go through all the terms and add them up one by one. But where's the fun in that? Let's take a more entertaining approach, shall we?
First, let's observe that the sequence seems to have a pattern. Each term is obtained by adding one and a half to the previous term, except for the first term which is 6. So, we can see that the sequence is formed by starting with 6 and then adding 1½, 1½, 1½, and so on.
Now, here comes the funny part. Did you know that summing arithmetic progressions is like herding clowns? Stay with me on this... Each term of this sequence is like a colorful clown stepping into a circus ring, and the sum we're looking for is like gathering all those clowns together!
To make things easier, let's call the first term "a" (which is 6), and the common difference between the terms "d" (which is 1½). So, the 13th term can be found using the formula: a + (n - 1)d.
Plugging in the values, we get: 6 + (13 - 1) × 1½.
Simplifying that gives us: 6 + 12 × 1½.
Now, to figure out the sum, all we need to do is perform a marvelous feat of clown herding known as the sum of an arithmetic progression. The formula for that is n/2 × (2a + (n - 1)d).
In our case, n (the number of terms) is 13, a (the first term) is 6, and d (the common difference) is 1½. Substitute those values and amusingly calculate the sum: 13/2 × (2 × 6 + (13 - 1) × 1½).
And the grand finale, after some math magic, reveals that the sum of the 13th term is... drumroll, please... 162! Ta-da!
Now, wasn't that a more thrilling way of finding the answer? I hope I brought a smile to your face, even if the clowns might make you a bit uneasy.
To find the sum of the 13th term in the given sequence, we first need to determine the pattern of the sequence so that we can find the formula to calculate any term.
From the given numbers: 6, 9, 13½, 21/4, we can observe the following pattern:
The first term is 6, the second term is 9 (which is 6 + 3), the third term is 13½ (which is 9 + 4½), and the fourth term is 21/4 (which is 13½ + 7/4).
From this pattern, we can observe that each term is obtained by adding a progressively increasing fraction to the previous term.
The fraction added starts with 3, then increases by a half each time. So:
The 1st term is 6
The 2nd term is 6 + 3 = 9
The 3rd term is 9 + (3 + 1/2) = 9 + 4 ½ = 13 ½
The 4th term is 13 ½ + (3 + 1/2 + 1/2) = 13 ½ + 7/4 = 21/4
So, the pattern is to add (3 + (n - 1)/2) to the (n-1)th term to obtain the nth term.
Using this pattern, we can then calculate the 13th term. Let's substitute n = 13 into the formula:
13th term = (13 - 1)/2 + 3 + 21/4
= 6 + 3 + 21/4
= 6 + 12/4 + 21/4
= 6 + 33/4
To add these two fractions, we need a common denominator, which is 4:
= (6 * 4)/4 + 33/4
= 24/4 + 33/4
= 57/4
Therefore, the 13th term in the sequence is 57/4.
Now that we have obtained the 13th term, we can find its sum by using the formula for the sum of an arithmetic sequence:
Sum of an arithmetic sequence = (n/2)(first term + last term)
Here n = 13, the first term = 6, and the last term = 57/4.
Plugging these values into the formula:
Sum = (13/2)(6 + 57/4)
= (13/2)(24/4 + 57/4)
= (13/2)(81/4)
= (13 * 81) / (2 * 4)
= 1053 / 8
Therefore, the sum of the 13th term in the given sequence is 1053/8.
To find the sum of the 13th term of the given sequence, we first need to determine the pattern of the sequence.
Looking at the sequence: 6, 9, 13½, 21/4, we can observe that each term is obtained by adding a constant difference to the previous term.
Let's break down the pattern:
- 6 + 3 = 9
- 9 + 4½ = 13½
- 13½ + 7/4 = 21/4
The difference between the terms is increasing by 1½ each time. So, we can find the nth term of the sequence using the formula:
nth term = first term + (n - 1) * difference
Now, let's calculate the 13th term of the sequence:
first term = 6
difference = 1½
n = 13
13th term = 6 + (13 - 1) * 1½
= 6 + 12 * 1½
= 6 + 12 * 3/2
= 6 + 18
= 24
Therefore, the 13th term of the sequence is 24.
To find the sum of the terms in an arithmetic sequence, we can use the formula:
sum = (n/2) * (first term + last term)
Now, let's calculate the sum of the 13 terms of the sequence:
first term = 6
last term = 24
n = 13
sum = (13/2) * (6 + 24)
= 6.5 * 30
= 195
Therefore, the sum of the 13 terms of the sequence is 195.