In a geometric series each term is 21, a statement that at first glance seems too simple to merit discussion, yet it opens the door to a deeper understanding of how geometric progressions behave when the common ratio equals one. In practice, this special case—where every term in the sequence is identical—illustrates fundamental concepts such as the role of the ratio, the behavior of partial sums, and the conditions for convergence. By unpacking what it means for a geometric series to have a constant term of 21, we reinforce the general formulas that govern all geometric series and see how they simplify under particular conditions. The following sections walk through the definition of a geometric series, derive the relevant formulas for this constant‑term scenario, examine finite and infinite sums, highlight practical applications, and address common points of confusion.
Understanding Geometric Series
A geometric series is a sequence of numbers in which each term after the first is obtained by multiplying the preceding term by a fixed, non‑zero number called the common ratio (denoted r). If the first term is a, the series can be written as
[ a,; ar,; ar^{2},; ar^{3},; \dots ,; ar^{n-1},; \dots ]
The n‑th term is therefore
[ T_n = ar^{,n-1}. ]
When we add the terms together we obtain the geometric series
[ S_n = a + ar + ar^{2} + \dots + ar^{n-1}. ]
Two important formulas arise from this definition:
- The n‑th term: (T_n = ar^{,n-1})
- The sum of the first n terms (when (r \neq 1)): (\displaystyle S_n = a\frac{1-r^{,n}}{1-r}).
If (|r| < 1) the infinite series converges, and its sum is
[ S_{\infty} = \frac{a}{1-r}. ]
These expressions hold for any geometric progression, regardless of the numeric values of a and r. The case where each term equals 21 simply fixes a and forces a particular value for r Most people skip this — try not to..
The Special Case: Each Term Equals 21
Saying “in a geometric series each term is 21” means that every term in the sequence has the same numeric value:
[ T_1 = T_2 = T_3 = \dots = 21. ]
Comparing this with the general term formula (T_n = ar^{,n-1}) we see that:
- The first term a must be 21, because (T_1 = a).
- For the second term to also be 21 we require (ar = 21). Substituting a = 21 gives (21r = 21), hence (r = 1).
- Checking the third term: (ar^{2} = 21 \times 1^{2} = 21), and so on.
Thus, the statement “each term is 21” uniquely determines a geometric series with
[ \boxed{a = 21 \quad \text{and} \quad r = 1}. ]
The common ratio being exactly one is the only way a geometric progression can produce a constant sequence. This observation is valuable because it highlights the boundary between convergent and divergent geometric series: when (|r| = 1) the usual convergence test fails, and we must treat the series separately.
Formula Derivation for the Constant‑Term Series
Plugging a = 21 and r = 1 into the general formulas yields interesting simplifications.
n‑th Term
[ T_n = 21 \times 1^{,n-1} = 21. ]
As expected, every term remains 21 regardless of n Took long enough..
Finite Sum
The standard sum formula (S_n = a\frac{1-r^{,n}}{1-r}) becomes indeterminate when r = 1 because both numerator and denominator approach zero. We resolve this by taking the limit as r → 1, or more directly by recognizing that adding n identical numbers simply multiplies the term by n:
[ S_n = \underbrace{21 + 21 + \dots + 21}_{n \text{ times}} = 21n. ]
We can also derive this from the limit:
[ \lim_{r\to 1} 21\frac{1-r^{,n}}{1-r} = 21 \lim_{r\to 1} \frac{-(n)r^{,n-1}}{-1} = 21n, ]
using L’Hôpital’s rule Small thing, real impact..
Infinite Sum
For an infinite geometric series we normally require (|r| < 1) to guarantee convergence. With r = 1 the series does not converge in the traditional sense; the partial sums grow without bound:
[ \lim_{n\to\infty} S_n = \lim_{n\to\infty} 21n = \infty. ]
Hence, the infinite series
[ 21 + 21 + 21 + \dots ]
diverges to infinity. If one insists on assigning a finite value via alternative summation methods (e.g., Cesàro summation), the result would still reflect the unbounded nature of the series, reinforcing that r = 1 sits at the edge of convergence.
Finite Sum Example
Suppose we want the sum of the first 15 terms of the series where each term equals 21.
Using (S_n = 21n):
[ S_{15} = 21 \times 15 = 315. ]
We can
