Question

In an A.P. if a4=84 and a10=60 then find the maximum possible partial sum of the sequence.

Answer 1

6d = -24, so d = -4

since d<0, the maximum partial sum is just
a1 = 84+12 = 96

Answer 2

actually, that is wrong.

when do the terms start going negative?
96 - 4n < 0
n > 24
so the greatest partial sum will be
S24 = 24/2 (96+0) = 1152

Answer 3

[AI-generated answer — Step-by-Step Bot]

To find the maximum possible partial sum of an arithmetic progression (A.P.), we need to determine the number of terms in the partial sum and the common difference between the terms.

Given:
a₄ = 84
a₁₀ = 60

We can find the common difference by subtracting the fourth term from the tenth term:

d = a₁₀ - a₄
d = 60 - 84
d = -24

Now that we know the common difference, we need to find the number of terms in the partial sum. We can determine this using the formula for the nth term of an arithmetic progression:

aₙ = a + (n - 1)d

Using the known values:
84 = a + (4 - 1)(-24)

84 = a - 72

a = 84 + 72
a = 156

Now we can calculate the number of terms in the partial sum using the formula for the nth term:

aₙ = 156 + (n - 1)(-24)

60 = 156 + (10 - 1)(-24)

60 = 156 + 9(-24)

60 = 156 - 216

-156 = -216

84 = (n - 1)(-24)

-2 = (n - 1)

n - 1 = -2

n = -2 + 1

n = -1

Since the number of terms cannot be negative, there must be a mistake in the given values or calculations.

Please verify the provided information.

Answer 4

[AI-generated answer — Explain Bot]

To find the maximum possible partial sum of the arithmetic progression (A.P.), we need to calculate the sum of the terms up to the maximum term. Here's how you can do it:

Step 1: Identify the common difference (d) of the A.P.
The common difference between consecutive terms in an arithmetic progression remains constant. To find the common difference (d), we can subtract a previous term from a later term. In this case, we can subtract a4 from a10.

d = a10 - a4
d = 60 - 84
d = -24

Step 2: Determine the term number (n) for the given terms.
Here, we know that a4 = 84 and a10 = 60. The term number is represented as n.

So, n1 = 4 and n2 = 10.

Step 3: Find the sum of terms up to n2.
The sum of terms up to n in an A.P. can be calculated using the formula:

Sn = (n/2) * [2a + (n - 1)d]

where Sn is the sum of terms, a is the first term, n is the term number, and d is the common difference.

Using the given values:

S10 = (10/2) * [2a + (10 - 1)d]
S10 = 5 * [2a + 9d]

Step 4: Substitute the given terms to find the maximum possible partial sum.
Substitute the values of a and d that we found in the previous steps:

S10 = 5 * [2(84) + 9(-24)]
S10 = 5 * [168 - 216]
S10 = 5 * (-48)
S10 = -240

So, the maximum possible partial sum of the sequence is -240.